ELEMENTARY   CALCULUS 


A  TEXT-BOOK  FOR  THE  USE  OF 
STUDENTS    IN    GENERAL    SCIENCE 


BY 


PERCEY   F.    SMITH,    Ph.D. 

// 

PROFESSOR  OF  MATHEMATICS  IN  THE  SHEFFIELD  SCIENTIFIC  SCHOOL 
OF  YALE  UNIVERSITY 


o>Ko 


NEW  YORK-:. CINCINNATI-:. CHICAGO 

AMERICAN    BOOK    COMPANY 


Copyright,  1902,  by 
PERCEY  F.  SMITH.' 

EL.  CALC.      SMITH. 
W.  P.   I 


PREFACE 

This  volume  has  been  written  in  response  to  the  un- 
mistakable and  growing  demand  for  a  text-book  on  the 
Calculus  which  shall  present  in  a  course  of  from  thirty- 
five  to  forty  exercises  the  fundamental  notions  of  this 
branch  of  mathematics.  In  American  technical  schools 
students  pursuing  courses  distinct  from  engineering 
branches  usually  terminate  their  mathematical  studies 
with  Plane  Analytic  Geometry.  But  in  view  of  the  recent 
remarkable  development  of  certain  of  the  general  sciences 
along  mathematical  lines,  such  a  course  can  no  longer  be 
regarded  as  adequate.  Moreover,  there  can  be  no  differ- 
ence of  opinion  as  to  the  relative  advantage  to  the  student 
of  a  knowledge  of  more  than  the  mere  elements  of  Ana- 
lytic Geometry  and  an  introductory  acquaintance  with  the 
Calculus.  It  is,  I  think,  the  experience  of  every  teacher 
that  the  average  student  first  realizes  the  power  and  use 
of  mathematics  when  taught  to  solve  problems  in  maxima 
and  minima  by  means  of  the  methods  of  the  Differential 
Calculus.  Certainly  no  stronger  argument  can  be  adduced 
in  favor  of  an  adjustment  of  the  curriculum  which  shall 
include  this  branch  of  mathematics.  Such  a  change  has 
been  effected  in  the  Sheffield  Scientific  School,  and  results 
abundantly  justify  the  step. 

For  the  general  student  in  our  colleges  who  elects  a 
year's  work  in  mathematics  beyond  the  usually  required 

3 


4  PREFACE 

Trigonometry,  the  most  satisfactory  course  would  seem  to 
be  one  in  which  the  time  is  equally  divided  between  Plane 
Analytic  Geometry  and  Calculus. 

In  writing  this  book  I  have  everywhere  emphasized  the 
possibility  of  appUcations.  The  examples  have  been  care- 
fully selected  with  this  end  in  view.  The  first  chapter 
may  seem  long,  but  the  notion  of  limit  certainly  demands 
adequate  treatment.  While  an  elementary  text-book  offers 
no  excuse  for  employment  of  the  refinements  of  modern 
rigor,  I  have  endeavored  to  avoid  positive  inaccuracies 
and  have  carefully  distinguished  between  demonstration 
and  illustration. 

I  am  indebted  to  my  colleague,  Dr.  W.  A.  Granville,  for 
many  helpful  suggestions. 

PERCEY  F.  SMITH. 

Sheffield  Scientific  School. 


CONTENTS 

CHAPTER 

I.     Functions  AND  Limits 7 

II.     Differentiation 23 

III.  Applications 5^ 

IV.  Integration 7° 

V.    Partial  Derivatives ^4 


ELEMENTARY   CALCULUS 

CHAPTER   I 

FUNCTIONS   AND    LIMITS 

1.  Continuous  Variation.  In  this  book  we  are  concerned 
with  real  ^lumbers  only.  Geometrically,  such  numbers  may 
be  conveniently  represented  by  points  of  a  scale  (Fig.  i). 


-f • 1 h- 


etc. -5    — i     —3—2-101234567   etc. 

Fig.  I 

Then  to  every  real  number  corresponds  one  point  of  the 
scale,  and  only  one;  conversely,  every  point  of  the  scale 
represents  a  real  number.  Any  segment  of  the  scale, 
however  small,  represents  indefinitely  many  numbers.  We 
speak  indifferently  of  the  number  a  and  the  point  a  of  the 
scale. 

A  variable  x  is  said  to  vary  continuously  between  the 
numbers  a  and  b  when  it  assumes  values  corresponding  to 
every  point  of  the  segment  ah, 

2.  Functions.  The  problems  arising  in  Elementary 
Calculus  involve  in  general  two  variables  in  such  a  way 
that  the  value  of  one  variable  can  be  calculated  as  soon  as 
a  value  is  assumed  for  the  other.  Thus,  in  Geometry,  the 
student  has  an  illustration  in  the  area  and  radius  of  a 
circle,  two  variables  such  that  the  area  A  can  be  calculated 
when  we  know  the  radius  r  from  the  formula  A  =  irr^. 

7 


8  FUNCTIONS   AND   LIMITS 

Definition.  A  variable  is  said  to  be  a  function  of  a  second 
variable  when  its  value  depends  upon  the  value  of  the  sec- 
ond variable  and  can  be  calculated  when  the  value  of  the 
second  variable  is  assumed. 

The  first  variable  is  called  the  dependent  variable,  and 
the  second  the  independent  variable. 

For  example,  the  equations 

J  =  ;r2,  J  =  sin;ir,  y  =  logio(;r2  -  i) 

state  that  j/  is  a  function  of  x.  In  the  first  two  cases,  j/ 
may  be  calculated  for  any  value  of  ;r ;  in  the  last  case,  how- 
ever, X  is  restricted  to  values  numerically  greater  than  i, 
since  the  logarithms  of  negative  numbers  cannot  be  found. 
In  the  first  two  cases,  then,  we  say  that  the  dependent 
variable  (or  the  function)  is  defined  for  every  value  of  x, 
and  in  the  last  the  function  is  defined  only  when  x  exceeds 
I  numerically. 

A  function  is  defined  for  a  value  of  the  variable  when  its 
value  can  be  calculated  for  that  value  of  the  variable. 

Elementary  Functions.  Pozver  Function:  x'^,  m  any 
positive  integer. 

Logarithmic  Function:  log«;r,  a>0]  this  function  is 
defined  only  for  x>o. 

Exponential  Function :  a"",  a>0,  i.e.  the  exponent  is  a 
variable,  the  number  a  being  a  constant. 

Circular  Functions  :'^  sin;r,  cosjr,  tan;r,  etc.,  i.e.  involving 
^       the  six  trigonometric  functions. 

*  So  called  from  the  use  of  the  circle  in  their  definition, 

e.g.  in  Fig.  2,  sin  AOP=  —  =  MP  if  P  is  the  unit  of 

P 
linear  measure.    Hereafter,  angles  will  always  be  meas- 

arc  *       y4  P 

ured  in  circular  measure,  i.e.  x  = =  — ^-.     In  the 

radius       R 
Fig.  2  unit  circle,  JR  =  i,  x  =  a.rc  A  P. 


FUNCTIONS   AND    LIMITS  9 

Inverse  Circular  Functions:  arc  sin  ;r,  arc  tan  ^',  etc.,  i.c, 
the  "arc  whose  sine  is  ;r,"  "arc  whose  tangent  is  a',"  etc. 
In  the  unit  circle  (see  Fig.  2)  /^  =  i  ;  if  x=  MP,  then 
arc  sin;ir=  arc^/*. 

One  thing  is  peculiar  here.  Assuming  any  value  of  x 
not  exceeding  i  numerically,  arc  sin  x  may  be  calculated, 
but  the  number  of  answers  is  always  indefinitely  great. 
For  not  only  is 

arc  sin  MP  =  arc  AP, 

but  also  equal  to  any  number  of  circumferences  4-  arc  AP, 

i.e,  arc  sin  MP  =  3,ycAP  +  2  irn, 

where  n  is  any  integer. 

For  this  reason  the  inverse  circular  functions  are  called 
many-valued  functions.  For  definiteness  we  may  always 
take  the  least  positive  arc. 

3.  Functional  Notation.  As  general  symbols  for  func- 
tions of  variables  we  use  the  notation 

A^\  ^iy\  H^l  etc., 

(read  /  function  of  x,  theta  function  of  y,  phi  function  of 
r,  etc.). 

We  mean  by  this  that  f(x)  is  a  variable  whose  value  de- 
pends upon  ,r,  and  can  be  found  when  a  value  is  assumed 
for  X.  The  notation  is  extremely  convenient,  for  it  enables 
us  to  indicate  the  value  of  the  function  corresponding  to 
any  value  of  the  variable  for  which  the-  function  is 
defined. 

Thus /(^)  represents  the  value  of /(,t')  for  x  =  a,  6(0) 
the  value  of  0  (y)  for  j  =  o,  (f>{-^)  the  value  of  <f){r)  for 
r  =  1-,  etc. 


lO  FUNCTIONS   AND   LIMITS 


EXERCISE  1 


1.    For  what  values   of  the   variable  are   the  following  functions 
defined  ? 

(a)  -.    Ans.     For  every  value  except  x  =  o,   since   -  cannot  be 
^  calculated.*  ° 


(J?)   y/x^  —  6x.     Since  x^  —  6x  or  x (x  —  6)  must  not  be  negative^ 
X2ivA  X  -  6  must  always  have  the  same  signs. 

Ans.     For  every  value  except  those  between  o  and  6. 
(0   ^jy  - y^ ;  W   ^lo;  W  arcsin;jr; 


(/)  arc  sec  ;r;  (g)  sin  Vi  -f  ;r;  (A)  logtan^r. 

2.  Given /(x)  =  x^  —  '/ x^+i6x— 12,  show  that /(2)  =o, /(3)=o. 
Does  /"(^')  vanish  for  any  other  value  of  ;ir  ? 

3.  Given  /(x)  =  logx]  show  that 

4.  Given  </>  (x)  =  a'  ;  show  that 

5.  Given  ^  (x)  =  cos  ;r; 

then  0(x)  +  6  (y)  ~  cos  ;r  +  cos/. 

From  Trigonometry,  we  know  that 

cos  X  +  cosy  =  2  cos  J  (^  +  /)  cos  J  (^  —  /)  ; 

4.  Graph  of  a  Function.  After  determining  for  what 
values  of  the  variable  a  given  function  is  defined,  it  is  im- 
portant to  know  in  what  manner  the  value  of  the  function 

*  The  student  should  observe  that  the  four  fundamental  operations  of  arithmetic, 
addition,  subtraction,  multiplication,  and  division,  v^^hen  performed  with  r<?a/ num- 
bers, give  real  numbers,  with  the  single  exception  that  division  by  zero  is  excluded. 


FUNCTIONS   AND   LIMITS  II 

changes  with  the  variable.  Geometrically  this  is  accom- 
plished by  drawing  the  graph  of  the  ftmctioHf  which  is 
thus  defined : 

The  graph  of  a  function  is  the  curve  passing  through  all 
poijits  whose  abscissas  are  the  values  of  the  variable  and 
ordinates  the  conrspondijtg  values  of  the  ftmction. 

In  the  language  of  Analytic  Geometry  the  graph 
of  a  function  f{x)  is  the  locus  of  the  equation 

y=f{x). 


I- 


-1- 


-2- 


X 1 


Fig.  3 

By  carefully  drawing  the  graph  of  a  func- 
tion a  good  idea  is  obtained  of  the  behavior 
of  the  function  as  the  variable  changes.     For  ex- 
ample, the  graph  of  logg.^,  i.e,  the  locus  of  the 
equation  * 

7  =  logs -^^ 
is  drawn  in  Fig.  3. 

Here  we  see  the  following  facts  clearly  pictured 
to  the  eye. 

{a)    For  ;r=  I,  logger  ^logg  I  =  O. 

{b)   For  x>\y  logg^  is  positive  and  increases  as  x  in- 
creases. 

♦  The  values  oiy  are  found  from  the  formula  proven  in  the  theory  of  logarithms, 
^'         logio  3 


12 


FUNCTIONS   AND   LIMITS 


(c)   For;r<i,  log^  x  is  negative  and  increases  indefi- 
nitely in  numerical  vahce  as  x  diminishes. 

{d)  For  X  =  o,  logg  X  is  not  defined,  since  the  logarithm 
of  zero  cannot  be  calculated. 

The  graph  of  the  general  logarithmic  function  log^  ;r 
may  be  drawn  by  merely  changing  the  ordinates  in  Fig.  3 
in  the  constant  ratio  i  -f-  logg  a. 

Graphs:    {a)  Of  :r2.  (^)  Of  ;ir8. 

F 


y 

j 

/ 

/ 

/ 

->  1 

,  ] 

~ 

r 

\   2', 

^ 

( 

a 

) 

~ 

y 

7-^ 

I 

r 

it 

-     -^ 

_     7 

^     ^ 

'      yr 

-^ 

X'                 4. 

'0  1^3                J^ 

H^J-- 

i 

t-      " 

J 

:        t^l 

-  -i  -% 

/ 

The  graph  of  x"^  has  the  appearance  of  {a)  or  (^)  according  as  tn 
is  even  or  odd. 


(c)   Of  logs  ;r. 


Z' 


-2 4^  =  =  , 


Ti 


id)  Of  3= 


2  3 


ffl 


X 


Since  if  we  set  y=  \og^x,   then  ;r=3J',  the  graph  in  (c)  has  the 
same  relation  to  XX^  and  YY^  as  (^)  to  KF'  and  XX K 


FUNCTIONS  AND   LIMITS 


13 


(e)  sin  .r. 


v-*^  ""*> 

"^SL-ia  52 

]_4.^. 

x'    -^^-    ^-i 

-^■■r                 K               J,C 

^^rr- 

1M_ 

\ 

3:     X' 


The  graphs  of  the  circular  functions  have  the  appearance  of  a  curve 
repeated  over  and  over  as  the  variable  increases  or  decreases.  As  in 
(c)  and  (d),  if  we  revolve  (e)  and  (/)  around  XX 'y  and  interchange 
XX'  and  VV,  we  shall  have  the  graphs  of  arc  sin  ,r  and  arc  tan  ,r 
respectively. 

5.  Limits.  For  the  study  of  the  Calculus  it  is  absolutely 
essential  that  the  student  should  understand  perfectly  the 
fundamental  notion  of  a  limit.  He  is  already  familiar  with 
simple  examples  of  limits  from  Geometry,  such  as  the  limit 
of  the  perimeter  of  an  inscribed  regular  polygon  as  the 
number  of  sides  is  indefinitely  increased  is  the  circum- 
ference, and  the  limit  of  the  area  of  the  polygon  is  the 
area  of  the  circle.  These  are  examples  of  variables  ap- 
proachijtg  limits,  the  variable  being  in  the  first  case  the 
perimeter,  and  in  the  second  the  area  of  the  regular 
polygon.  The  following  definition  states  the  matter  gen- 
erally. 


Definition.     A  variable  is  said  to  approach  a  number  A 
as  a  limit  when  the  values  of  the  variable  ultimately  differ 


14  FUNCTIONS   AND   LIMITS 

from  A  by  a  number  whose  numerical  value  is  less  than 
any  assignable  positive  number. 

If  we  represent  the  values  of  the  variable  by  the  infinite 
sequence 

a^y  ^2'  ^3'  *">     ^wj  ^n+i)  •*•> 

then  on  the  scale  (Fig.  i)  the  points  corresponding  to 
^1,  ^2'  ^3'  ***'  ^ny  ^n+\i  *"y  ^tc,  will  ultimately  approach 
nearer  the  point  A  than  any  assignable  length,  that  is,  will 
*'heap  up"  at  the  point  A.     The  definition  interpreted 

A 


I 
Fig.  4 


<—h- 


-h-^ 


geometrically  means,  then,  that  no  length  h  (Fig.  4),  however 
small,  can  be  laid  off  from  the  point  A,  but  that  points  of 
the  sequence  will  fall  within  the  segment. 

We  write  Limit  {a^  =  A,  or,  also,  if  we  denote  the  varia- 
ble whose  values  are  a^,  a^,  etc.,  by  x, 

Limit  (;r)=^. 

6.   Limiting  Value  of  a  Function.    Continuous  Function. 

Consider  the  elementary  function  log^^r  (Fig.  3).     Take 

any  sequence 

a^  a^y  a^y  •••, 

of  positive  numbers  whose  limit  is  some  positive  number^. 
For  example,  the  sequence 

1.3,  1.33,   1.333,  -, 

the  limit  of  which  is  |-.  Consider  now  the  sequence  of 
numbers 

loga(^l),     ^Oga{a^)y     l0ga(^3),   ••', 

and  draw  their  ordinates  in    Fig.  3.     Then  the  student 


FUNCTIONS   AND   LIMITS  1 5 

will  see  that  this  last  sequence  has  the  Hmit  \oga{A); 
that  is, 

IV/ien  the  variable  x  approacJies  a  limit  A  greater  than 
zero^  the  logarithmic  function  log^x  approaches  the  limit 
logaA. 

We  express  this  important  fact  by  writing 

Limit  (loga^r)^^  =  loga^. 

The  general  relation  brought  out  by  this  example  is  the 
following:  When  the  values  assumed  by  the  variable  x 
approach  *  a  limiting  value  A,  then  the  corresponding 
values  of  the  function  will  also  approach  a  limiting  value ; 
and  if  the  function  is  defined  for  the  value  A,  then  the 
limiting  value  of  the  function  is  its  value  for  x=^  A.  Or, 
in  symbols,  \i  f{A)  is  a  number,  then 

Umit{f{x)),=,=f(A). 

For  example,  since  cos  0=1,  Limit  (cos  ;i')^=o  =  i  •  The 
property  above  described  is  that  of  contiftuity ;  i,e.  a  cofi- 
tiniwiis  function  is  such  that 

Limit /(;r)  =/(  Limit  x). 

For  the  purposes  of  the  Calculus  it  is  essential  that  a 
function  should  be  continuous.  The  elementary  functions 
of  §  2  possess  this  property. 

7.  Infinity.  If  the  points  on  the  scale  of  Fig.  i  cor- 
responding to  the  sequence  of  values  of  the  variable  x 

*  The  variable  x  may  approach  the  limit  A  in  any  manner  consistent  with  the 
definition  of  the  function.     In  the  above  illustration  the  geometrical  sequence 

I.  i  +  i.   i+J  +  iV.   i  +  i  +  A +A,etc., 

whose  limit  is  §,  might  also  have  been  taken. 


l6  FUNCTIONS   AND   LIMITS 

ultimately  advance  to  the  right  without  Hmit,  we  say,  ";r 
increases  without  limit,"  or  also,  '' x  approaches  the  limit 
positive  infinity,"  and  we  write 

Limit  ;i:  =  -f-  oo. 

If  under  the  same  conditions  the  points  advance  to  the 
left  without  limit,  we  say,  '^;ir  decreases  without  limit,"  and 
write 

Limit  ;r  =  —  00. 

Finally,  if  the  points  advance  both  to  the  right  and  left 
without  limit,  we  write 

Limit  ;ir=  00. 

The  student  should  disabuse  his  mind  of  any  previous 
notions  of  infinity  not  agreeing  with  the  above  definitions. 
The  symbols  +  ^,  —  ^,  ^,  must  be  used  always  in  the 
sense  above  described. 

8.  Fundamental  Theorems  on  Limits.  The  student  is 
asked  to  accept  the  following  theorems  as  true : 

Given  a  number  of  variables  whose  limits  are  known ; 
then 

I.  The  limit  of  an  algebraic  sum  of  any  finite  number 
of  variables  equals  the  same  algebraic  sum  of  their  respective 
limits. 

II.  The  limit  of  the  product  of  any  finite  number  of 
variables  equals  the  product  of  their  respective  limits, 

III.  The  limit  of  a  quotient  of  two  variables  equals  the 
quotient  of  their  respective  limits  when  the  limit  of  the 
denominator  is  not  zero. 


FUNCTIONS   AND   LIMITS 


17 


9.   Two  Important  Limits.     To  prove 
Limit 


rsinxl 
L    a?    Ja5=0 


In  Fig.  5  let  ;r=arc  A  T 
=  arc  AS,  the  radius  OQ 
being  taken  equal  to  unity. 
Then 

tan;r=  TQ  ^  QS. 
Now 


Fig.  5 


ST<^rcST<SQ  +  QT, 
.'.  2  sin  ^ <  2  ^ <  2  tan  x ; 
whence,  dividing  through  by  2  sin  jr, 

I 


T  <    ^     ^  tan.r 


sin  ;ir      sin  ;r  \      COS  x 
Therefore,  taking  reciprocals, 


C0S;r<5Hl^<I. 

X 


;) 


Now  let  ;r  approach  zero  as  a  limit ;  then,  since  cos  0=1, 

id  t 
have 


sin  X 
and  the  value  of lies  between  i  and  cos^,  we  must 


_  .    .    fsm  x^ 

Limit    =  I . 

L   ^   Jx=o 

10.    Consider  next  the  infinite  sequence 
I  I  I.I 


♦  Since  for  ;r  =  o,  ^^^^ 
not  defined  for  x  =  o,      ^ 
EL.  CALC.  —  2 


=  -,  a  meaningless  expression,  the  function is 


i8 


FUNCTIONS    AND    LIMITS 


Representing  the  successive  terms  by  a^,  a^,  a^,  ...,  we 

have 

=  I 

=  2 

=  2.5 

=  2.666... 


«1  = 

I, 

«2  = 

i+i 

«3  = 

i  +  - 

^4  = 

i+- 

an  = 

i+- 

numbers  of 

+  - 


I  -2' 


+A+-  ^ 


1-2        1-2.3 

+  .    \    ,  +  -'  +1—^*,  etc. 


1-2        I  •  2  •  3 


/^  —  I 


this  sequence  continually  increase.     We 
may  show,  however,  that  any  term  is  less  than  3. 
For    \r_ >  2^*,  and  therefore 

I 
I       I       I        I                  I               2^ 
^»<^t7  +  o  +  :r2  +  :73  +  -+:;^=^+-; > 


I     22^ 


— - 1 


since 


.1  +  1  +  1  +  .     ._L 

2  ^  22  ^  2^  ^         ^  2^- 


is  a  geometrical  progression  and  its  sum  may  be  imme- 
diately written  by  the  usual  formula. 

Hence  <2„  <  3 -—^  and  taking  ;?  =  i,  2,  3,  etc.  ad  infi- 

nitunty  every  term  of  the  sequence  is  seen  to  be  less  than  3. 

6 

^1 0^2  a  3      I 


r 


Fig.  6 


T 


The  points,  then,  corresponding  to  the  sequence  (Fig.  6) 
must  heap  up  at  some  point  to  the  left  of  3 ;  that  is,  the 
sequence  must  have  a  limit. 

♦  The  symbol  \n  —  i,  read  "  factorial  n  —  i,"  means  the  product  of  all  integers 
from  I  to  «  —  I  inclusive. 


FUNCTIONS   AND    LIMITS 


19 


The  calculation  of  this  Hmit  to  any  number  of  decimal 
places  is  a  matter  of  no  difficulty,  as  the  following  compu- 
tation to  five  decimal  places  will  show. 


Write  down 
Divide  by 


t,-' 


-2_ 


2^^ 


Adding, 


i.oooooo(=  i). 
2)1.000000  (  =  — J 

3  ). 500000  (^=^^ 


4).  166667  (^=,| 


5).04i667(^=.-ij 
6).oo8333  (^=^ 
7).ooi388(=|) 


8).oooi98 


'll^ 


9).oooo25  (^=|^j 
io).ooooo3  (=r^ 


2.71828 


neglecting  the  figure  in  the  sixth  decimal  place,  of  which 
we  cannot  be  sure,     In  fact,  it  can  be  easily  shown  that 


20  FUNCTIONS   AND    LIMITS 

2.71828  is  the  limit  of  the  sequence  correct  to  five  deci- 
mal places. 

Writing  the  limit  of  the  sequence  in  the  form  of  an  infi- 
nite series  and  denoting  this  Umit  by  ^,  we  have 

e  =  l+i  +  i  +  i  +  ,^H-i+  etc.,  ad  infinitum. 

1     [2     [3     [i     L» 

6  =  2.71828.... 

The  number  e  is  called  in  the  Theory  of  Logarithms  the 
Napierian  base  or  natural  basey  and  is  a  number  of  prime 
importance  in  mathematics. 

The  expression  for  e  in  the  form  of  an  infinite  series 
should  be  remembered  and  also  its  value  to  five  decimal 
places. 

11.   To  prove 

Limit  ri+-T      =e. 

A  rigorous  proof  of  this  very  important  limit  is  beyond 
the  scope  of  this  volume.  We  may  perhaps  best  illustrate 
the  meaning  of  the  theorem  by  drawing  the  graph  of  the 
function  for  positive  values  of  ^. 


Setting  J/  =  f  I  +  -  j  ,  then 

logioJ  =  ^logio(^i+^), 


and  for  any  value  of  ;s  greater  than  zero  y  may  be  approxi- 
mately calculated,  as  for  example  in  the  accompanying 
table,  which  gives  j/  to  five  decimal  places. 


FUNCTIONS   AND 

LIMITS 

z 

y 

.01 

1.04723 

.1 

1.27098 

I. 

2. 

10 

2.59374 

100 

2.70481 

1000 

2.71692 

10,000 

2.7I8I5 

100,000 

2.71827 

1,000,000 

2.71828 

21 


etc. 


Fig.  7 


The  figure  illustrates  the  theorem  in  showing  that  the 
graph  approaches  the  Hne  j  =  ^  as  z  increases  indefinitely. 
When  z  diminishes  toward  zero,  y  approaches  unity.  . 


22  FUNCTIONS   AND   LIMITS 

EXERCISE  2 

[The  graph  of  the  function  considered  must  be  drawn  in  every  case."] 

1.  Prove  Limit  r-^^-3-^+4]      ^  2. 

L        x-i        JiB=2 

We  have  merely  to  substitute  2  for  x. 

2.  Prove  Limit  P^  ~  ^^1       =-2a. 

We  cannot  substitute  directly,  for  we  should  get  -,  a  meaningless 

x^  —  a^  o 

expression.     But  =  x  —  a^  and  we  may  now  substitute. 

x+  a 

3.  Prove  the  following  in  which  a  is  any  number  greater  than  zero  : 

Limit  [—1      =  +  00;    Limit  f-l      =00: 

Limit  \ci^\      =  00 ;    Limit    -         =0. 

*-        —'35=00  '-'^— 'x=ao 

The  last  three  results  are  often  written 

a  a 

-  =  00,  ^  •  00  =  00,   —  =  o, 

O  00 

but  the  student  must  remember  that  such  equations  are  merely  abbrevia- 
tions of  the  preceding. 


4.    Prove  Limit  f  "^^'+^  -  ^1      =  _L 

L  h  Jft^     oa/ 


Hint.     Multiply  numerator  and  denominator  by  y/x  +  h  -\-  Vx. 

5.    Show  that 

Limit  r^^l       =1;    Limit  [tan  :r1       =00; 
Lsm;rJ;c^  L  Jx=f 

Limit    logeX        =  -  00  ;    Limit  f^"*!       =  o. 


CHAPTER   II 


DIFFERENTIATION 


12.  Increments.  In  order  to  understand  the  manner 
of  variation  of  a  function  as  the  variable  varies,  it  is  essen- 
tial to  know  how  great  a  change  in  value  occurs  in  the 
function  for  a  given  change  in  value  of  the  variable. 
Change  in  value  is  termed  mcrement ;  i.e.  the  incremetit  of 
the  function  is  the  change  in  value  of  the  function  corre- 
sponding to  a  given  change  in  value  or  increment  of  the 
variable. 

The  problem  now  arises  :   To  calculate  the  increment  of  a 

given  f  miction. 

Let  f{x)  be  defined  for 
all  values  of  x  from  x  to 
x  +  h  (Fig.  8).  Now  for 
X  -{-h  the  value  of  the  func- 
tion is  f{x  +  h)j  hence  the 
increment  of  the  function 
fix)  corresponding  to  an 
-li-A  A      hicreme7it  h  in  the  variable 


-hr^ 


X 

Fig.  8 


J{x+h)-f(x) 


a?+/i 


X  ts 


f{x  +  h)-f{x). 


We  shall  represent  the  increment  of  any  variable  by  the 
letter  A  (read  **  delta  ")  prefixed  to  that  variable,  thus 

If    Ax  =  ^,  then  A/(ir)  =  /(ir  +  7^)-/(x). 

23 


24  DIFFERENTIATION 

Rule.  To  find  the  increment  of  a  functiony  calculate  the 
new  value  of  the  function  by  replacing  x  by  x  ■\-  h  and  sub- 
tract the  old  value  of  the  function  from  the  new  value, 

EXERCISE  3 

1.  Find  t:^\  t^'^  ={x  ^  hy  -  x^=:2kx-\-  h^,    Ans. 

2.  Find   AfiV  a(1)='-1=—Z±-.  Ans, 

\xl  \xl     x  +  h     X     x{x  +  h) 

3.  Prove    A>/r=  ' (Ex.  4,  page  22.) 

y/x  +  -^  +  Vx 

4.  Find     Alogjr. 

Alog;r=log(;ir+>^)-log;ir=:logf  ^^^ j  =  log  f  I  +-J.     Ans. 

5.  Find    A  sin  x. 

A  sin  X  =  sin  {x-\-  K)  —  sin  x—7.  cos  {x^\K)  sin  \  h, 

from  Trigonometry.     Ans. 

6.  Find    A^.  A^  =  e^+^  -  e"^  =  e'^^e^  -  i).    Ans, 

7.  Find     A  cos  2  x.  A  cos  2X=—  2  sin  (2  ;ir  +  h)  sin  h.     Ans. 

8.  Find    t^W+x.  ,  ^ns. 

2VI  +  X 

13.  The  Increment  Quotient.  While  the  increment  of 
a  function  as  found  in  the  preceding  article  is  of  impor- 
tance, still  more  essential  in  any  investigation  is  the  rate 
of  change  of  the  function,  that  is,  the  change  iit  the  function 
per  unit  change  in  the  variable. 

If  we  form  the  quotient 

A/(^) 


t=0 


DIFFERENTIATION  25 

we  obtain  the  average  rate  of  change  of  the  function  while 
the  variable  changes  from  x  \o  x  -\-  h. 

For  example,  the  **  law  of  falling  bodies," 
given  in  Mechanics,  asserts  that  the  distance  s 
traversed  by  such  a  body  falling  freely  from  rest 
in  a  vacuum  varies  as  the  square  of  the  time  /, 
that  is, 

the  constant  16.1  being  determined  experimen- 

^^2    tally  when  s  is  measured  in  feet  and  t  in  seconds. 

Therefore  A^  =  16.1  (^  +  lif  -  16.1  i^. 

As 
or  — =  16.1(2 /-f //),  since  At  =  k 

At 

For  example,  the  average  velocity  throughout 

As 
the  third  second  is  given  by  setting  in  —^t  =  2, 

Fig.  9      A  =  I,  and  is  80.5  feet  per  second. 

EXAMPLES 

1.   From  Physics  we  learn  that  for  a  perfect  gas  at  constant  tem- 
perature the  product  of  the  pressure  p  and  volume  v  is  constant,  or 

pv  =  2i  constant  c,  i.e.  p  =  -  ;  show  that     —  —  — 


UZ 


V '  Av         v^  +  vAv 

2.  Show  from  Ohm's  law,  viz.  current  strength  C  equals  electro- 
motive force  E  divided  by  the  resistance  R^  that  for  constant  R  the 
change  of  current  strength  per  unit  change  of  electromotive  force  is 
constantly  equal  to  i  -^  R. 

14.  Derivative  of  a  Function.  In  the  illustration  taken 
from  the  law  of  faUing  bodies  given  in  §  13,  let  us  propose 
to  ourselves  to  find  the  velocity  at  the  end  of  two  seconds. 
Making  /=  2,  we  have 

^  =  64.44-  16. 1//, 
At 


26 


DIFFERENTIATION 


which  gives  us  the  average  velocity  throughout  any  time 
h  after  two  seconds  of  falhng.  Our  notion  of  velocity 
shows  us,  however,  that  by  the  velocity  at  the  end  of  two 
seconds  we  do  not  mean  the  average  velocity  during  one 
second  after  that  moment,  or  even  during  y^Q-  or  -joVcr 
of  a  second  after  that  moment,  but,  in  fact,  we  mean  the 
limit  of  the  average  velocity  when  h  diminishes  toward 
zero;  that  is,  the  velocity  at  the  end  of  two  seconds  is 
64.4  feet  per  second.  Thus,  even  the  everyday  notion 
of  velocity  involves  mathematically  the  notion  of  a  limit, 
or,  in  our  notation. 

Velocity  =  Limit  f^l 

Thus,  after  t  seconds  have  elapsed,  the  velocity  of  a 
faUing  body  is  32.2  t  feet  per  second. 

Again,  let  it  be  required  to  find  the  slope  of  the  tangent 
at  any  point  P  of  a  plane  curve  whose  equation  is  given 
in  rectangular  coordinates  x  and  y. 


The  tangent  at  P  is  constructed  as  follows  (Fig.  10) : 
Through  P  and  any  point  P^  on  the  curve  near  P  draw 
the  secant  AB.     Let  the  point  P^  move  along  the  curve 


DIFFERENTIATION  2^ 

toward  P,  the  secant  AB  meanwhile  turning  around  P. 
Then  when  P^  coincides  with  P,  the  secant  AB  becomes  -^ 
the  tangent  TP, 

Now  if  P  is  (x,  y)  and  P^  {x  -h  A;r,  y  +  Ay),  the  slope  of 
yi^is 

,      ^      SP^      ^y 
tan^=— -  =  --^- 

As  P'  approaches  P  as  above  described,  £^  will  ap- 
proach zero  as  a  limit,  while  0  approaches  the  angle  PTO 
or  7 ;  hence,  at  the  limit, 

tan  7  =  Limit  [ -r^  ]         =  slope  of  tangent  at  any  point  JP, 

\Aa?/Aj5=o 

For  example,  the  slope  of  the  tangent  at  any  point  of 
the  parabola  J  =  x^  -f-  3  is  2;ir. 

Law  of  Linear  Expansion.  \i  Iq  is  the  length  of  a  rod  at  o°  Centi- 
grade, and  /  the  length  at  t°  on  the  same  scale,  then  experiment  estab- 
lishes the  law  of  expansion 

1  =  Iq  +  at  +  bt\ 

a  and  b  being  constants.    The  coefficient  of  linear  expansion  at  any  tem- 
perature /  is  the  increase  in  length  per  unit  change  in  temperature,  i.e. 

coefficient  of  expansion  =  Limit  ( — ) 

We  easily  find,  then,  that 

coefficient  of  expansion  =  a  +  2  bt, 

and  .'.  a  =  the  coefficient  of  expansion  at  o^. 

Specific  Heat  of  a  Substance.  The  specific  heat  of  any  substance  is 
the  quantity  of  heat  necessary  to  raise  a  unit  mass  of  the  substance 
one  degree  in  temperature.  If  Q  is  the  measure  of  the  quantity  of  heat 
in  unit  mass,  and  /  the  corresponding  temperature,  then  by  definition, 

specific  heat  —  Limit  (  — ^  ) 


28  DIFFERENTIATION 

These  examples  show  that  we  obtain  an  important  new 
function  of  the  variable  if  we  can  find  the  limit  of  the 
Increment  Quotient  when  the  increment  of  the  variable 
approaches  zero.  This  function  is  called  the  derivative 
of  the  function. 

Definition.  The  derivative  of  a  function  is  the  limit  of  the 
quotient  of  the  increment  of  the  function  and  the  increment  of  the 
variable  when  the  latter  increment  approaches  the  limit  zero. 

The  step  of  finding  the  limit  of  -^^  ^  when  A;r  ap- 
proaches  o  is  indicated  by  changing  the  A's  to  ordinary 
^'s,  so  that  -^—-^  =  Limit  (    -{  '  ^  |       ,  or  also,  if 

ax  \    1\X    /Aa'=0 

A^  =  ,»,    g§^  =  Limit  r^("+\>-^^^>l       . 

The  symbol    -^^  ^   is    read,    "derivative   of  f{x)  with 
ax 

respect  to  xT     This,  being  a  new  function  of  x,  is  often 
written  /'  {x\  so  that  also, 


%^=/'(- 


dx 
Thus  in  the  illustrations  given, 


)• 


velocity  =  — , 

i.e.  velocity  is  the  derivative  of  the  space  traversed  in  the 
time  t  with  respect  to  the  time. 

Slope  of  tangent  =  -^, 
dx 


DIFFERENTIATION  29 

i.e.  equals  the  derivative  of  the  ordinate  of  the  poi7it  ivith 
respect  to  its  abscissa. 

Coefficient  of  linear  expansion  =  — , 

dt 

or  the  derivative  of  the  length  with  respect  to  the  teviperature. 

Specific  heat  =  -^, 
^  dt 

that  is,  equals  the  derivative  of  the  quajitity  of  heat  in  unit  mass  with 
respect  to  the  temperature. 

Many  more  illustrations  of  physical  magnitudes  might  be  given  which 
take  the  form  of  a  derivative. 

We  call  —  the  sign  of  differentiation^  so  that  the  pre- 
dx 

fixing  of  —  to  any  function  of  x  means  that  the  following 
dx 

process  is  to  be  carried  through : 

General  Rule  of  Differentiation.  1°.  Calculate  the 
qnotient  of  the  increment  of  the  function  and  the  increment  of  the 
yariable  (i.e.  the  increment  quotient). 

2°.  Find  the  limit  of  this  quotient  when  the  increment  of  the 
variable  approaches  the  limit  zero.* 

It  must  be  emphasized  here  that  the  characteristic  thing 
in  differentiation  is  finding  the  limit  of  a  quotient.  From 
the  standpoint  of  the  Differential  Calculus  a  function  is  of 
no  interest  if  the  limit  mentioned  does  not  exist.  Func- 
tions possessing  derivatives  are  said  to  be  differoitiablcy 
and  it  is  of  prime  importance  to  show,  for  example,  that 
the  elementary  functions  of  §  2  are  differentiable. 

♦  The  student  must  notice  that  the  limit  of  the  increment  quotient  cannot  be 
found  by  Theorem  III,  §  8,  since  the  limit  of  the  denominator  is  zero. 


30  DIFFERENTIATION 

15.   Differentiation  of  the  Elementary  Functions 
oc^j  sin  Xf  log^  00, 

(a)  To  prove  — x^^  =  ^nx""^'^. 

ax 

Now  A  {x^)  =^{x-\-  lif  -  ;r"^  if  A;ir  =  //. 

But  {x  +  hy  =  X'''  +  ;;/;ir^- V^  +  •  •  •  +  /i"\ 

the  terms  not  written  containing  powers  of  //. 
.-.  A(x'^)  =  mx"'-^/i  +  ...  +  //^ ; 

A;r 
where  again  the  terms  not  written  contain  powers  of  //. 
Putting  ^  =  o,  we  find 

(i)  4-(^"')  =  ^^'^~^' 

ax 

(b)  To  prove  —  sin  x  =  cos  x. 

dx 

Since  A  sin  ;r  =  sin  (x  -^  h)  —  sin  x 

=  2  cos(;r+ J>^)  sin^/^  (§  I2,  Ex.  5), 
we  find 

A  sin  ;r  __  2  cos  {x  •\-\K)  sin  \  h 
Ax  h 

/     ,   1  ,x    sinii^ 
=  cos  {x  +  1^)  •  — r|~* 

Limit  (5i^)      =1'  (§9), 

and  Limit  (cos  {x-\-\  /i))h=o  =  cos  x 

(since  cos  x  is  continuous,  §  6), 
so  that  we  may  apply  the  theorem  II,  §  8,  and  we  have 

(2)  -—  sin;ir=  cos;r. 

ax 


But 


DIFFERENTIATION 


31 


(c)  To  prove  — -  log^  x  =  log^  e  — 
From  §  12,  Ex.  4,  we  have 


I  +-f 

xj 


since  the  introduction  of  the  exponent  -  is,  by  the  princi- 

h 

pies  of  logarithms,  equivalent  to  multiplying  the  logarithm 
itself  by  that  exponent. 

Now  f  I  +  -  j^  is  the  expression  of  §  11  if  we  write  in  that 


expression  ^  =  -■• 
h 


Also 
hence     Limit 


imit[| 


Limit 


A=0 


I  +  ^"jfl     =  Limit 

xJ     1/1=0 


.-.  Limit  flog^  (^1+^)1 


i+- 


=  l0ga^, 


and  we  have 

(3) 


(since  log  ;ir  is  a  continuous  function), 

ax  X 

Formula  (3)  becomes  most  simple  when  ^  =  ^,  for  then 

d  ,  I 

—  log,;r  =  -. 
dx  X 

Logarithms  to  the  base  e  are  called  natural  logarithms  or  Napierian 
logarithms  (§  10),  and  the  factor  logo^  in  (3)  is  called  the  vwdubis  of 
the  system  whose  base  is  dr,  i.e.  the  7iu?nber  bywhich  natural  logariih77is 
fntist  be  7nultiplied  in  order  to  obtain  logarithjns  to  any  given  base  a. 

We  write  M  =  modulus  =  loga  e. 


32 


DIFFERENTIATION 


For  example,  the  modulus  of  the  common  system  of  base  lo  is  log^)  Cy 

and 

logio^  =  0.43429 
to  five  decimal  places. 

If  in  (i),  (2),  and  (3)  we  write  u  for  x^  we  have 

(4)     ^u^'^  =  mu^-^'^    -^sinii=cost/j    ^  loga  «^  =  loga  ^ -• 
^^     du  du  du  u 


EXERCISE 

4 

1.   Diiferentiate  with  respect  to  x. 

(a)  x^-]-sax+d. 

Ans. 

2x-\-3a. 

(^)  .%■ 

Ans. 

a 

{pc^by 

(c)    Vx  (cf.  Ex.  4,  p.  22) 

Ans. 

I 

2.  Prove  —  cos  :r  =  —  sin  x. 

dx 

3.  Prove  — vT+^= — 

dx  2V1  +;r 


2^/x 


4.    Prove  ^(^^V- ■^• 


5.    Prove  — (O/)  =  C,  if  C  is  any  constant. 
du 


6.   From  the  law  of 

falling  bodies 

we  found  (§14) 

1  =  3..-. 

or 

velocity  =  -z/  =  32.2 1, 

Prove 

f-3- 

What  does 
represent  in  Mechanics 

?                                            Acceleration. 

Ans, 

DIFFERENTIATION  33 

7.  Find  the  velocity  and  acceleration  of  the  motion  defined  by 
(i)  s  =  at  -\-  ^g/'^;  Ans.  V=a-\'gt\  accel.=^. 
(2)  s  =  at  -  \gt'^'^                    Ans.   V=a—gt'^  accel.=  — ^. 

8.  Find  the  slope  of  the  tangent  to  y  =  6x—  x'^  at  the  origin. 

Ans.  6. 

16.  Certain  General  Rules.  We  prove  in  this  section 
several  important  rules  for  differentiation  of  a  general 
character. 

Let  the  variables  //,  v,  w,  etc.,  be  functions  of  the  vari- 
able X. 

I.    To  differentiate  any  algebraic  sum  of  these  variables. 

For  example,  to  find  — -  iti  -\- v  —  w). 
ax 

Now 

=  b^u  -\-  Lv  —  Azv, 

A(7i  +  z^  —  '^)  _  A?/      Av     Aw 
Ax  Ax     Ax     Ax 

Since     Limit  M       =  $^,     Limit  M       =f', 

r  .    .^rAzvl  dzv 

Limit   — -         =  -— , 

LA,rjA:r=()         dX 

we  may  apply  I,  §  8,  and  we  have 

/^.  d  ,      ,  ^      du  ,   dv      dzv 

IL    To  differentiate  a  product. 

For  example,  to  ^et  -r-{uv). 
dx 


34  DIFFERENTIATION 

Now  A  (uv)  =  (^  +  A^)  {v  +  J^v)  —  tWy 

=  ^  Az^  +  V  An  +  A^/  Av, 

A(tcv)        Av  ,      A?/  ,    .     Av 
Ax  Ax        Ax  Ax 

Since     Limit  r^l       =  ^,     Limit  f^l       =^, 
LA;trjAa:=o      ax  y_Axj>^x^     ax 

Limit  [A^]^^o=o, 

we  may  apply  I  and  II,  §  8,  and  obtain 

/^N  d  y     ^         dv  ,      dti 

(6)  ^^^'^)=^'^  +  '^^- 

To  find  -— -  {uvw\  consider  tivw  as  made  up  of  the  two 
dx 

factors  uv  and  w ;  then,  by  (6), 

(uv  'W)=UV f-  w  -^ ^, 

dx  dx  dx 

or  by  (6)  again, 

(7)  =  //z^ h  Z£//^  -—  +  ^^  — 

dx  dx  dx 

III.    7!:?  differentiate  a  quotient. 

Q*      A  (^^  _u  -\-  Au  _u  __  vAu  —  uAv 
\vj     V  +  Av      V        v'^  +  v  Av 

Au        Av 

V u 

A  AA__  Ax        Ax 

Ax\vj        v^-}-vAv 

Then  since   Limit  [z^2  +  ^Az/]^^o= '^^  we  may  apply  I, 
II,  III,  §  8,  and  have 

dv        du 
(^)  d   /u\         dx        dx 


dx  \v 


DIFFERENTIATION  35 

From  (6),  (7),  and  (8)  we  have  the  rules : 

I.  The  derivative  of  an  algebraic  sum  of  any  number  of  vari- 
ables is  equal  to  tlie  same  algebraic  sum  of  the  derivatives  of  the 
variables. 

II.  The  derivative  of  a  product  of  any  number  of  variables  is 
equal  to  the  sum  of  all  the  products  formed  by  multiplying  the 
derivative  of  each  variable  by  all  the  remaining  variables. 

III.  The  derivative  of  a  quotient  equals  the  denominator  times 
the  derivative  of  the  numerator  minus  the  numerator  times  the 
derivative  of  the  denominator,  all  divided  by  the  square  of  the 
denominator. 

To  these  we  may  add  the  following : 

IV.  The  derivative  of  a  constant  is  zero. 

y.  The  derivative  of  a  constant  times  a  variable  equals  the 
constant  times  the  derivative  of  the  variable. 

Rule  V  comes  from  (6)  and  IV,  if  we  place  u  equal  to  a 
constant. 

EXAMPLES 
1.    Workout  -^(H-;r2)(i -2;f2). 

Rule  II  is  first  applied,  and  we  get 

^(i-f;f2)(i-2;f2)  =  (i  +  ;f2)  ^(i  _2;r2) +  (1-2^2)^  (i  +;r2). 
ax  ax  ax 

By  Rule  I,         £  0  — ^)  =  ^  (0  -  £(-^ 

Since  by  V,  —  (2:^2)  =  2  —x\  and  from  (4)  ^  15,  — ^2  =  24-, 
dx  dx  '  dx 

we  have  finally, 

^  (I  +  ;r2) (I  - 2 :r2)  =  (i  +;r2)  .  -4^+  (l  -2 :i'2)  .  2  ^=  - 2 x{\  +  4-^'^)- 


36  DIFFERENTIATION 


d  f  sinx\ 
dx\\ogexl 


2.   Work  out  . 

Vlog, 

Rule  III  we  use  first  and  find 

,    ,    .  loge X-—  sin  or  -  sin x  —-  loge x 

a    I  sin  -^  \  _  dx  dx 

dx   \  loge  ^')  ~  (loge  X)  ^ 

By  (4)  §15?  —  sin ;r=cos^,  —loge :r=i. 

d  /  sin .^'N  _  ;r cos .r loge ;tr— sin ;r 
dx  \  loge  xi  X  (loge  ^)  ^ 

EXERCISE  5 

Prove  the  following  differentiations  : 

■yd.          \      ,      ^                       ^     d  /sin;r\      ;ircos;ir— sinjr 
1.   — x(\  —  X)=\  ~-2X.  3.    —     = 

dx    ^  ^  dx\    X    I  x'^ 

2     ^  (     ^     \=     ^  ~^^  4    i^/L±j£!^  =  __4£__. 

'dxKi+xy    (i+x^y  'dx\i-xy    (i~x^y 

5     ^  (    ^"*    \  —  ^^'^0^^  +  x^  {in  —  n)) 
dx\l  +;ir«/  ~  (I  +;ir")2 

6.   — (;ir"'log:r)  =  jr"»-^(i  +  ;//log;r). 
dx 

'  dx\i  -xV       (I  -;t-2)2' 

'  dx\aJ      a  '  dxW'l      ;r"+i 

10.  —  :r"*(i  -:r)"  =  ;i"»-i(l  -xy-^{m-  {7n  +  n)x), 
dx 

Special  attention  should  be  given  to  the  following: 

1-1     T--   J  d  .  c"         i.  sin  11 

11.  Find  — tan?/.      Since  tan  2/ = , 

du  cos  u 

,  d   .  d  fsin7i\ 

we  have  —  tan  ?/=---    • 

du  dii\cQS,ul 

Applying  III  (4),  §  15,  and  Example  2,  §  15,  we  find 

—  tan  2/  =  sec^?/. 
du 


DIFFERENTIATION  37 


Prove 


12.  —  cot  u  =  —  cosec-  21. 
du 

13 


.  —  sec  ti  =  sec  u  tan  //.      (  Put    sec  71  — ] 

du  \  cos  //  / 

14.  —  CSC  7i  =  —  CSC  u  cot  H. 
du 

17.    We  come  now  to  two  most  important  rules. 
Differentiation  of  Inverse  Functions.     Suppose  j  is   a 
function  of  x^  i.e.  in  symbols 

(9)  y  =fi^)- 

Then  it  is  usually  possible  inversely  to  calculate  x  when 
values  are  assumed  for  j,  i.e.  we  may  choose  y  for  the 
independent  variable  instead  of  x,  so  that  by  solving  (i) 
for  X  we  obtain 

(10)  ;r  =  (/)(;/). 

Then  f{x)  and  <t>{y)  are  called  inverse  functions, 

Exa?nple.     If  y  =  a',   then  x  =  loga/ ;   that  is,  a^  and   loga/  are 
inverse  functions. 

Let  now  Ax  and  Aj/  be  corresponding  increments  of  x 

and  J/,  so  that  Ax  and  Ay  vanish  together,  since  we  are 

dealing  here  with  continuous  functions.     Then  the  incre- 

.     Ay         Ax  ,.  .    .  1 

ment  quotient  is   -^  or  -r—,  according  as  x  or  y  is  taken 

for  the  independent  variable. 

Now  by  multiplication,   -r^  .  x— =  i, 

hence  Limit! -r^j        .  Limit f-r^)        =1, 


38  DIFFERENTIATION 

by  II,  §  8,  since,  as  above  emphasized,  Aj/  and  Ax  vanish 
together.     We  have,  therefore,  in  the  derivative  notation, 

dy    dx  .  . 

^.^=i,orsolvmg, 

(II)  '^  =  i-* 

dx 

VI.  If  1/  is  a  function  of  oc^  and  inversely  oc  a  function  of  2/,  then 
the  deriratiye  of  x  with  respect  to  y.  equals  the  reciprocal  of  the 
derivatire  of  y  with  respect  to  x. 

Differentiation  of  a  Function  of  a  Function.  We  have 
seen  by  (4)  how  to  differentiate  with  respect  to  x  the  ele- 
mentary function  sin  x.     Suppose  we  wish  to  find 

-^sin(l+;ir2), 
dx 

for  which  the  rule  (4)  does  not  suffice.  We  then  introduce 
the  variable  u=  i  -^x'^y  and  setting;^  =  sin(i  '{■x^)=^  sin^, 
we  have  before  us  the  relations 

(12)  jK  =  sin^,    ^  =  I  4-^^ 

and  we  say  y  is  a  function  of  x  through  u^  i.e.  y  is  z.  function 
of  a  function. 

Now,  if  Aj,  A?/,  and  Ax  are  corresponding  increments  of 

y,  u,  and  x,  then  forming  the  increment  quotients  -=^,  — -, 
we  have,  by  multiplication, 

(\%\  ^     AM^Ay^ 

^  ^^  Au  '  Ax     Ax 

♦The  student  will  not  fail  to  notice  that  in  (ii)  the  familiar  property  of  a 
fraction,  -=  i  -^  -  is  suggested.    But  he  must  not  forget  that  -~-  is  noi  a  fraction, 

Ay 

but  merely  the  symbol  for  the  limiting  value  of  the  fraction  ^. 


DIFFERENTIATION 


39 


But  the  increments  A;/,  Au,  and  Ax  vanish  together,  so 
that,  by  II,  §  8, 

Limit  ( -r^  J        •  Limit  f^)        =  Limit  f-j^) 

\AuJ^u=0  \^'^Jax=0  \AxJ^^^ 

or  (lA)  dy^dydu^ 

^  ^^  dx     clu    ddc 

VII.  If  2/  is  a  function  of  oc  tlirough  w,  then  tlie  derivative  of  y 
with  respect  to  a?  eqnals  the  products  of  the  derivatives  of  y  with 
respect  to  u  and  of  u  with  respect  to  cc. 


Thus   in  (12),  since    — sin//: 
au 


cos//,    -7-  (i  +  ^^)  =  2r,  we  find 
ax 


-7-  sin  (i  +  x'^^  =  cos  /^  •  2  ;ir  =  2  :r  cos  ( i  +  x^) . 
dx       ^  "^ 

EXERCISE  6 

1.  Show  that  the  geometrical  significance  of  (i  i)  is  that  the  tangent 
makes  complementary  angles  with  XX'  and  W. 

2.  If  a  material  point  /*,  whose  rectangular  coordinates  are  x  and  /, 
move  in  a  plane,  then  x  and  y  are  functions  of  the  time  /.     Now  the 


horizontal  component  v^  (see  Fig.  11)  of  the  velocity  v  is  the  velocity 
along  OX  of  the  projection  M  of  /*,  and  is  therefore  the  time  rate  of 


40  DIFFERENTIATION 

change  of  ;r,  or  -z/j  ==    - .     In  the  same  manner,  the  vertical  component 
^2  equals  ^;  and  since 

we  have 


"V{f)V(f)'. 

For  the  direction  of  the  velocity,  tan  y  =  --,  or 

dy     dx 
tan  y  =  -f-  -f-  -— . 
^      dt      dt 

3.  Prove  that  the  equations  x  —  a  cos/,  y  —  a  sin/  define  uniform 
motion  in  the  circle  x'^  -\- y^  =  a^. 

4.  If  the  coordinates  (^, /)  of  a  point  on  a  curve  are  functions  of  a 
variable  ^,  show  that 

(At\  dy  _(iy  ^dx 

^  ^^  dx~  dO     dO' 

(Use  (14)  and  then  (u).) 

d      ^ 

5.  To  find  — (x'i)  when  {/  is  any  positive  integer;  i.e.  to  differ- 

dx 

entiate  arty  root  of  x. 
1 
Put  //  —  x'i,  then  x  =  W^  ]    hence,  by  (4),  §  15, 

dx 


and  by  (11) 


dn         I 


dx     qufi"^ 


But  n'i~'^  —  X  "i    =  X    9, 


...  ^-^  =  1;^'/  \  or  ^{x<i)-^-x~^  '; 
dx     q  dx  q 


i.e.  ///<?  same  ride  holds  for  roots  and  powers  of  the  independent 
variable. 


DIFFERENTIATION 


41 


6.    From  (4),  §  15,  and  VII,  we  have 


liv^     ^     <///^       lit-  dx 


(16) 


du 


dx  du  dx  dx 


du 


r/  ,  </     ,  .du       ,  dx 

-^log«//  =-r-(log<./0;77.=  log..— 


dx 


du 


7.    To  find  ^(-r*). 

I 
Letting  //  =  .v*  (Example  3),  we  have 


..  ^  (.r«)  =^u'=  puP-^  —  ( Example  6) 


dv 


t:^^     I    \-i      p    e_, 


//^«cv  the  rule  of  (4),  §  15.  for  pinvers  holds  for  «/; 

du 

8.    To  prove    .arc  sm  //  =     . 

^  dx  Vi  -  u^ 

Placing/  =  arc  sin  //,  we  have  inversely, 

//  =  sin/. 


\    {.  ( 'fi  in, 


du 
dy 


-  cos/,  and  by  (15), 


^  _    J 

du       COS/ 


But 
Hence 

and  by  (16), 


COS/  =  Vi  —  sin'-*/  =  Vi  —  «^. 
^=:        '       , 

dy      d  .  dx 

;  =  -J-  arc  Sin  «  =    , 
dx     dx  Vi  -  «a 


ensurabU 


42  DIFFERENTIATION 


d 

dx 

du 
dx 

I  +«2 

9. 

Prove 

arc  tan  u  = 

(Remember  sec^j/  =  i  + 

tanVO 

10. 

Prove 

d 

dx 

arc  sec  u  = 

du 
dx 

uy/u'^  —  I 

11. 

Prove 

d 
dx' 

V'  =  a"*  loge 

dx 

Putting  y  -. 

=  ^« 

,  we  have  inversely, 

u  =  ^ogay. 

du 
"  dy- 

=  l0ga^|    (§15, 

(4)). 

...  ^. 

_    y      _    a^   ^ 

du     loga  e     loga  e 

dx     dx  dx 

In  particular,  -^  ^«  ^  ^u  du. 

dx  dx 

Example  ii  shows  that  the  exponential  function  e"^  possesses  the 
remarkable  property  of  being  its  own  derivative^  for 

dx  dx 

In  general,  if  a  is  any  constant,  then 

(i )  -- ^«^  =  ^^«^,  since  —  {ax)  =  a ; 

dx  dx 

that  is,  the  derivative  of  the  function  ^"*  is  proportional  to  {i.e.  a  times) 
the  function  itself.  For  a  reason  now  to  be  explained,  the  function  ^«^ 
is  said  to  follow  the  Compound  Interest  Law, 

If  P  dollars  be  drawing  compound  interest  at  r  per  cent,  then  in  the 

time  t^t  the  interest  is Pt^t.  and  hence  the  change  in  P  or  A/*  is 

given  by 

£^P  =  —  P^t,  or  ^=I-P 
loo  A/      loo 

*  From  the  principle  in  logarithms,  loge  a  =  — - — • 

logatf 


DIFFERENTIATION  43 

Now  suppose  the  interest  to  be  added  on  contimiously^  and  not  after 
finite  intervals  of  time  A/,  i.e.  we  make  A/  approach  the  limit  zero,  and 
conceive  of  P  as  increasing  continuously ;  then 

dt      100 

so  that  a  sum  of  money  accumulating  continuously  at  compound  inter- 
est has  precisely  the  property  above  enunciated  in  (i),  viz.  its  derivative 
is  proportio7ial  to  the  sum  itself. 

18.  From  the  examples  in  Exercises  5  and  6  and  the 
Rule  VII,  we  deduce  the  following  fundamental  formulae 
for  differentiation : 

VIII.  -^  tiw*  =  mum-x  ^  {rn  any  commensnrable  number) . 

CfwK/  (toe 

du 
r^      d   .  -  dx 

X.  -^ aw  =  aw  logrc  a^(a  any  positiye  constant), 
ciQiy  doc 

XL   -^  sin  w  =  cos  1^  4^. 
dx  doc 

XII.  -^costi  =  -sinM^. 
doc  dx 

XIII.  -^tanu  =  sec2M^. 
dx  dx 

XIV.  -^  cot  u  =  -  csc2  u  ^. 
dx  dx 

XV.  — -  sec  li  =  sec  w  tan  w  — • 
dx  dx 

XVI.    -^CSCU=-C8CWC0tW^. 

dx  dx 

du 

XVII.  #  arc  sin  u  = —J^. 
dx  Vl  - 1*2 

du 
XVIII.  #arccosti=         ^^ 


da?  Vl  -  u2 


44 


DIFFERENTIATION 


du 
XIX.  #  arc  tan  ^/  =  ^^^ 


doc  1  +  w2 

XX.  -^  arc  cot  «^  -       ^  ^ 


XXI.  -^arcsecti- 


du 
XXII.  #arccsct«  =  -       ^^ 


^/a?  u  Vu^  -  1 

The  formulae  and  rules  I-XXII  the  student  must  memo- 
rize. With  their  aid  differentiation  of  the  commoner  func- 
tions is  made  rapid  and  easy,  but  perfect  famiharity  with 
them  is  indispensable. 

To  show  the  application  of  the  rules  three  examples  are 
now  given : 

1.   Find     ^f-J—^\ 

By  III,  ^^-^\= ^ -^ ^ 

By  I  and  IV,     —  (i  -:r)=:-i; 
dx 

from  VIII,  —  (I  +  x'^y  =  -(I  +  ;r2)"i^  (I  +  x^), 

dx  2  dx 

and  since  —  (i  +  ^^)  =  2  ;r,  we  have 

dx 


To  simplify,  muUiply  numerator  and  denominator  by 
(I  +  x'')k 


2^~^  . 


DIFFERENTIATION  45 

Then,  since         (i  -f-  x'^)^  =  i,  we  have,  reducing,' 


^Av^T+T;;       (n-;.^)* 


2.    Find     ^logeV'"'^^'- 
^-i-  ^  I  +  cos  ^ 


P'or  convenience,  set         y  =  log«  '\— 


Since       \oge^i'--^^=l\og{l  -  cosx)-Uog{i  ^  cosx), 
then  by  I  and  V, 

Applying  IX,  we  have 

,  — (i-cosa-)  (i+cosjir) 

dy      I  dx  I  dx  J  ,     VTT 

-^  —  -,  and  by  XII, 


dx     2      I  —  cos  .r  21+  cos  x 

^  _  I  /     sin  A-  sin  x     \  _      sin  .r      _     i 

^^  ~  2  V I  —  cos  ^     I  +  cos  .r/      I  -  cos'-^  ^  ~  sin ; 


^  1 ^/i  —  cos:r        I 


+  cos  ;r     sin  x 


3.   Find     ^^arctan(l^i:!). 

Setting  the  function  equal  to/,  we  have,  by  XIX, 
d  (e^  -  e-^\  e^  +  e  ^ 


dy  _   de\      2       .    _ 

dS  /e^j-j-ey-  ^^e2e  _  2  +  ^  2^  K^Y  ^) 


'  eo  ^  e-9 


46  DIFFERENTIATION 

EXERCISE  7 
Prove  the  following  diiFerentiations  : 


2.  aJl±^=—-1— 

dx^i  -X     (I  -x)^/T~- 
3.   Af_^\= I . 


4.    ^  /    3x^  +  2 


+  i)t) 


5.  ^(I  -2X+  3X^-  4x^)(i  -^  xV=-  20;i'3(i  +;ir). 

6.  —(I  -  3  ;ir2  +  6;ir4)(i  +  ar2)8  =  6o;r5(i  +  ;r2)2. 
dx 

7.  ^  (5x^2*)  =  2(ar  +  I)  5"='+^ log*  5- 
ax 

8.  — -  x'^a^  =  ;»r*»-^^*(«  4-  ^loge  a) . 
dx 

9.  ^  r£l2Il£  +  log.(i  _  :,r)l  =     1°S'^   . 

dxV  \-x        ^^         ^J     (I  -xy 

10.  -fl^(;^s_i£?  +  6£_6\^  ^^,^_ 
^;r      \  a         d^      a^J 

11.  /iog,(.x  +  .-)  =  ^^^i:!. 

12.  ^(VJ_log,(i+VJ))  = ?_^. 

13.  -^  tan2  5  ^  =  10  tan  5  ^  sec2  5  6. 
du 

14.  4h  si^^  OcosO  =  sin2  ^  (3  cos2  ^  -  sin^  0) . 

15.  -^  log  sec  ^  =  tan  ^. 


DIFFERENTIATION  47 

16.  40an2^-logsec2^)=2tan8^. 
dd 

17 .  -^  sin  «^  sin'*  0  =  n  sin"-^  ^  sin  («  +  i )  0, 
dd 

18.  — -  arc  sin  (3  ;r  -  4  ;r3)  =  — -^ . 

19.  ^arcsec-  =  -        ^ 


dx  a      x\/x^  -  d^ 

20.   -^  arc  esc        '        -        ^ 


^;r  2  ;r2  —  I       Vi  —  x^ 

21.  ^  arc  sin'- ^'-  ^ 


^  1+^2  I  +  ;r' 

22.   ^  arc  tan  .^+'^-       ' 


^;r  I  —  ^AT      I  +  ;r2 

23.   --  arc  cos  -  — 


^jr  e^  -{■  e-'     ef  -\-  e- 


A— 


24.  —arc  sec-'    ^     -  ^ 


^/r  ^\-\-x     2V1  —  ;r2 

25.  ^ (arc cot ^  + log. A^)^    ^''^^ 


x^-a'^ 


19.  Differentiation  of  Implicit  Functions.  If  an  analytic 
relation  is  given  between  two  variables  not  solved  for  either 
variable  in  terms  of  the  other,  then  either  variable  is  said 
to  be  an  implicit  ficnction  of  the  other. 

For  example,  in  x^  —  y^  -\-(^  =  o  either  x  ox  y\^  an  im- 
plicit function  of  the  other  variable. 

In  such  a  case  either  variable  may  be  chosen  for  the 
independent  variable,  and  if  we  can  solve  explicitly  for  the 
other  (as  in  the  above  example  for  y,  giving  y  =  V^^  —  9), 
then  we  can  differentiate  as  before.  But  it  is  generally 
better  not  to  solve  the  equation,  but  to  differentiate  the 
given  relation  as  it  stands. 


48  DIFFERENTIATION 


Thus,  to  find  -J-  from 

x'^  —  2)Xy  -\-  2/2  —  2. 

^  _  2  ;ir—  3/ 


and 


^;ir     3^— 4>' 


To  justify  this  process  is  beyond  the  Hmits  of  this  text- 
book. One  thing  is  to  be  noted,  namely,  that  only  those 
values  of  the  variables  which  satisfy  the  original  relation 
can  be  substituted  for  the  derivative. 


EXERCISE   8 

Fi 

dv 
nd   ~  from  the  following  equations  : 

1. 

y'^  -  1  xy  -  a'^. 

Aits. 

dy  _     y 
dx    y  —  X 

2. 

;t'2      1/2 

Ans. 

dy   .,     b\x' 
dx~     ay' 

3. 

ax'^  +  2bxy  +  cy^  +  2 

\fx-\-  2gy  +  // 

—  o. 
Ans. 

dy  _     ax+  by  +  /^ 
dx~     bx  -^  cy  -^  g 

4. 

x^  -{.  y^  —  ^  axy  =  o. 

Ans. 

dy  _     x'^  -  ay 
dx~     j2  _  ^x 

5. 

;rt  +  J/3   zz:  ^¥. 

Ans. 

dy         y^ 
dx        ^\' 

6.  Given  r  =  «(i  —  cos  ^)  ;   show   -^=^sin^. 

du 

n     r^-  9         9  n       I,         dr      —  <22  sin  2  ^ 

7.  Given  ^2  =  ^2  QQs  2  ^ ;    show   — -  =  — . 


DIFFERENTIATION  49 

20.  Derivatives  of  Higher  Orders.  Since  the  derivative 
of  a  function  of  a  variable  x  with  respect  to  x  is  also  in 
general  a  function  of  ;r,  we  may  differentiate  the  derivative 
itself,  that  is,  carry  out  the  operation, 

This  double  operation  is  indicated  by  the  more  compact 
notation, 

and  this  new  function  is  called  the  second  derivative.     In 
the  same  way, 

is  the  third  derivative^  and  in  general, 

is  the  «th  derivative  of  f(x\  that  is,  the  result  of  differen- 
tiating f{x)  n  times.     The  following  notation  is  also  used, 

|/(.r)=/'(:r),   g/(.-)=/"(-r),  ...,    £./(^)  =/(«)(^). 

The  operation  of  finding  the  successive  derivatives  is 
called  successive  differejttiation. 


EXERCISE  9 

1.   Given 

/(x)=    3,r^  -  4^-^  +  6 A' - 

-  I 

then 

/'(r)=  i2;i-8-8:i'+6: 
/"(:r)=  36:1-2 -8,  etc. 

EL.  CALC.  —  4 

50 


2.  Given 

3.  Given 

4.  Given 

5.  Given 

6.  Given 

7.  Given 


DIFFERENTIATION 

f{x)  =€'"'' ;  prove  /("^(;r)  =  ^«^«=". 
f{x)  =  loge  (I  -  X)  ;  prove  /(«)(:r)  = 


J  =  x^\ogeX\  prove 


^_6 
^;r*  ~  X 


(i  -  r)" 


_y  =  loge  sin  ;ir;  find 


^3j/  ^  2  cos  AT 
^;irS  ""  sin^  x  " 


y  =  e^''(x^-SX+s);   find  ^  =  8;fV*. 


jr-^     jK 


^2  ^^2 
From  Ex.  2,  Exercise  8, 

dx~ 


+•^2  =  i^  or  ^'^^^  +  ^y  =  ^^^^  to  find 


^Ar2  a4y2  ^ 


^2^' 


ay  dx 

dx'^'  aY  ' 

then  substituting  for  -^  and  reducing, 

dy  _     b\a^yl^  b^x'^)  _       b^ 
dx^  ~  i^p  ~     aY 

9.   From    y'^-zxy=a\  prove  ^  =  ^_-^^. 


.t. 

xy 


CHAPTER   III 
APPLICATIONS 

21.  Tangent  and  Normal.  For  all  applications  of  the 
Calculus  to  Geometry  the  fact  established  in  §  14  is  of 
fundamental  importance,  viz. 

Theorem.  TJie  value  of  the  derivative  of  y  with  respect 
to  X  found  from  the  equation  of  a  curve  in  rectangular  coor- 
dinates gives  the  slope  of  the  tangent  at  any  point  on  that 
curvcy  or 

—  =  slope  of  tangent. 
dx 

If  we  wish  the  slope  at  any  particular  point  (x\y\  we 
have  to  substitute  xf  and  ^  respectively  for  x  and  y  in  the 

general  expression  for  -^-      Let  \-^\    be  the  value  of 
,  dx  \dxj 

-^  after  this  substitution,  then  we  have  from  Analytic 
dx 

Geometry, 

Equation  of  the  tangent  at  (x\  y')  is 

(17)         ^--^' =(!)'(---')• 

Since  the  normal  is  perpendicular  to  the  tangent,  and 
Equation  of  the  normal  at  {x\  y')  is 

(18)  ^_y  =  _(^gy(^  _;,'). 


52 


APPLICATIONS 


EXERCISE  10 

1.  Find  equations  of  tangent  and  normal  to  the  parabola  j^  =  4^1- _[.  i 
at  the  point  whose  ordinate  is  3. 

Substituting  3  for/,  we  find  x  =  2,  hence  (x',  y')  is  (2,  3).    Differen- 

Ans.  tangent,  2  ;i'  —  3  j  +  5  =  o  ;  normal,  3  ;r  +  2/  —  12  =  0. 

2.  Find  equation  of  tangent  to  the  ellipse  b'^x^  +  a^y'^  =  aW  at 
(x',y').  Ans.  d^x'x  +  a^'y  =  aP-U^. 

3.  Show  (Fig.  12)  that  the  subtangent  M^T^  =  -/'(  — V,  and  the 
subnor7nal  M^N^  =  yH-^\  • 

4.  Prove  that  the  subnormal  in  the  parabola  j/^  =  ^px  has  the  con- 
stant length  2  p. 

22.  Sign  of  the  Derivative.  An  important  question  is 
the  following : 

Is  the  function  vtcrcasijig  or  decreasing  as  the  variable 
passes  through  a  given  value  a  f 

The  phrase  "  passing  through  a  "  is  understood  to  mean 
that  the  series  of   values  assumed   by  the  variable  is  an 


Fig.  12 


increasing  sequence  including  a,  i.e.  on  the  graph  of  the 
function  we  proceed  from  left  to  right.     In  Fig.  12,  as  we 


APPLICATIONS  53 

pass  through  P^  the  ordinates  are  decreasing,  while  at  P^ 
the  ordinates  are  increasing,  and  since  the  ordinates  repre- 
sent the  values  of  the  function  and  -^  or /'(a-)  is  the  slope 

ax 

of  the  tangent,  we  have  the  result : 

The  functioji  f{x)  is  increasing  or  decreasing  as  x  passes 
through  a  accoj^ding  as  f{a)  is  greater  or  less  than  zero. 

At  P^  and  P^  (Fig.  12)  the  tangent  is  parallel  to  XX' ^  and  therefore 
f'{x)  vanishes  at  these  points.  For  such  values  of  .r,  therefore,  the 
rule  just  given  does  not  enable  us  to  answer  the  question  proposed. 

If,  now,  for  any  value  of  ;r,  say  x=a,  the  second  deriva- 
tive ~^y  or  f"{x\  is  positive,  then  as  x  passes  through  a^ 
dx^ 

the  first  derivative  f\x)y  or  tan  7,  must  be  an  increasing 
function  of  x,  i.e.  7  must  be  increasing  as  x  passes  through 
a ;  and  therefore  as  we  pass  along  the  curve  from  left  to 
right,  the  tangent  is  rotating  counter-clockzvise,  and  the 
curve  is  accordingly  concave  upward  {diS  at  {a),  Fig.  13). 


Fig.  13 

On  the  contrary,  \{  f"{a)<o,  the  reasoning  shows  the 
tangent  to  be  rotating  clockivise  as  we  pass  along  the  graph 
through  X  =  ay  and  hence  the  curve  is  concave  downward 
Un  Fig.  13). 


54  APPLICATIONS 

The  result  is : 

A  curve  is  concave  upward  or  downward  as  x  passes 

through  a  according  as  the  value  of  the  second  derivative 

dh  * 

-y^  for  X  =  a  is  greater  or  less  than  zero, 

dh 
As  before,  if  -f^  =  o,  the  rule  just  given  does  not  enable 

d^y 
us  to  decide.     If  -f^  =  o  for  x=  a  and  changes  sign  as  x 

passes  through  ^,  then  at  x=  a  we  have  a  point  of  inflec- 
tion (P4  and  Pg,  Fig.  12). 

EXERCISE  11 

1.  Show  that  the  following  functions  are  either  always  increasing  or 
always  decreasing,  and  draw  the  graphs  in  each  case : 

.(^)tan^;     (d)  e-;     (c)  logx',     (^)  i. 

2.  Show  that  jk  =  sin;r  has  a  point  of  inflection  at  each  intersection 
with  XX'. 

3.  Determine  the  points  of  inflection  of  y  =(x  —  ay  -\-  d. 

Ans.  {a,  b). 

23.  Maxima  and  Minima.  A  function /(;r)  is  said  to  be 
a  maximum  for  x=^a  when  f(a)  is  the  greatest  value  of 
f{x^  as  X  passes  through  a. 

A  function  f{x)  is  said  to  be  a  minimum  for  x  =^  a  when 
f{a)  is  the  least  value  of  f{x)  as  x  passes  through  a. 

In  other  words,  a  maximum  value  is  greater  than  any 
other  in  the  immediate  vicinity,  and  similarly  for  a  mini- 
mum value.  It  is  not  to  be  inferred  that  a  maximum  value 
is  the  greatest  of  all  values  of  the  function ;  on  the  con- 
trary, a  function  may  have  several  maxima. 


APPLICATIONS 


55 


Graphically,  at  a  maximum  we  have  a  highest  point 
(Pj  and  /^3,  Fig.  14),  at  a  minimum  a  lowest  point 
(7^2  and  P4). 


Fig.  14 


Since,  by  definition,  if  f{d)  is  a  maximum,  f{x)  must  be 
an  ina^easing  function  f or  ;r  <  ^  and  a  decreasing  function 
for  jr  >  ^,  we  have  (§  22) : 

Theorem.  If  f{a)  is  a  maximum  value  of  f(x),  then 
the  first  derivative  f\x)  must  change  sign  from  positive  to 
negative  as  x  passes  through  a. 

By  similar  reasoning  for  a  minimimt,  we  find  a  change 
in  sign  from  negative  to  positive  micst  occur  in  fix). 

In  either  case,  therefore,  f'{x)  must  change  sign.  If  we 
now  assume  that  f'{x)  is  continuous  for  ;r  =  (^,  we  see  that 
f\a)  =  o ;  that  is,  the  tangent  at  a  highest  or  lowest  point 
must  be  horizontal  {P^  and  P^  in  Fig.  14).  If,  on  the  con- 
trary, f'{x)  is  not  continuous  for  x  =  a,  then  the  change  in 
sign  occurs  by  passage  through  00  ;  i.e.  the  tangent  becomes 
parallel  to  FF',  as  at  P^  and  P^.  This  case  is,  however, 
of  minor  importance,  and  is  omitted  from  further  con- 
sideration. 

Furthermore,  if  f"(a)<Of  the  curve  at  .r  =  ^  is  concave 
downward,  and  we  have  a  highest  point  {Pi),  while 
f"{a)>o  indicates  a  lowest  point  {P^^^ 


56 


APPLICATIONS 


We  have  therefore  the  following 

Rule  for  determination  of  Maximum  and  Minimum 
values  of  a  function  f{x). 

Find  the  first  derivative  f  {x),  and  get  the  roots  of  the 
equation  f{x)  =  0. 

First  Test.  If  f  (x)  changes  sign  as  x  passes  through 
any  root  a  of  the  equation  ^'(jr)  =  0,  then  f{a)  is  a  maxi- 
mum or  minimum  value  according  as  the  change  is  from  + 
to  — ,  or  from  —  to  +. 

Second  Test.  Find  the  second  derivative  f  (jr) ;  then, 
if  a  is  any  root  of  f'{x)  =  0,  f{a)  is  a  maximum  if  f"(a)  <  0, 
and  a  minimum  if  f  (a)>0.  If,  however,  f'(a)  =  0,  we 
must  use  the  first  test. 

EXAMPLES 

1.  Examine  the  function  ;i'3  — 3  jr^  —  9  jr  +  5  for  maxima  and  minima. 
Placing  /(x)=x^—  ^x^—  gx+  ^, 

then  /'  (x)  =^x^—  6x—g, 

and  the  roots  of  3;ir2— 6;ir— 9  =  0  ^ive  x=  2  and  —  i . 

Now    f'(x)  =  6x-6,  and /"(3)=i2,  f'(-i)=-i2y 
hence  by  the  Rule,  Second  Test, 

/(S)  —  —  12  is  a  minimu7ii  value, 
and  /"(—  i)=ioisa  maxi7nuin  value  of  the  function. 

The  student  should  draw  the  graph. 

2.  Examine  the  function  ^^  ~" — ^  for  maxima  and  minima. 

Here  /(;r)  =  ^-^~  ^^'- 

Differentiating  and  reducing,  we  find 


APPLICATIONS  57 

The  roots  of/'  (jr)  =  o  are  therefore  x=  i,  x=  5.     We  now  apply  the 
First  Test,  since  it  is  unwise  to  form  the  second  derivative. 
Taking  account  of  the  signs  only,  we  have 

When*  .r<  i,  f(x)  =  -  ^~)(~l  =-   ] 

•^  ^  ^  +  I      Hence  /{x)  is  a 

When   x>  I,  r(x)  =  -  (  +  )(-)  =  +      ^^^inimum  when  x=  i. 

When  x<s,/'ix)=-  C  +  K")  =+    ) 

+  I       Hence  /'  (;f)  is  a         ^  ^ 

When  x>  5,  /'(,r)  =  -  i±i(±l  =  -  J  ^^^^-^^'^^^^^^^^  when  ^'  =  5.    (g; 

.-.  y(i)  =  o  is  a  minimum,  and  /is)T  'kj^  maximum  value  of  the 
function. 

EXERCISE   12 

1.    Examine  the  following  functions  for  maxima  and  minima : 

(a)  x'^  —  3x-\-  5.  Minimum  value( 1 1 .^  Ans. 

(b)    -•  Max.  value  \,  min.  value  -  \.  Ans. 

I  -\-x^ 

{c)    6^+3  X-—  4x^.  Max.  value  5,  min.  value  —  J.  Ans. 

(ci)  x^—  3  .r2+  6x,  No  max.  or  min.  values.  Ans. 

(e)    ax'--i-2_l)^  -^  If  ^>o,  min.  value  '^^  "  ^  ^  if  ^<o, 

a 

then is  a  maximum.  Ans. 

a 


(/)    \oyJ%x-x^.  Max.  value  40.  .4;/^. 

This  function  is  a  maximum  or  minimum  according  as  8  x  —  x'^  is 
a  maximum  or  minimum,  hence  f  ^  constant  factor  or  a  radical  sign 
may  be  dropped  in  investigations  of  this  sort. 

*  We  consider  values  of  x  differing  only  very  slightly  from  the  number  on 
the  right  of  the  inequality  sign. 

t  If  w  is  any  polynomial  in  .v  containing  no  multiple  /actors,  we  may  show  that 
\lu  is  a  maximum  or  minimum  only  when  «  is  a  maximum  or  minimum.     For  if 


58  APPLICATIONS 

2.  Divide  the  number  a  into  two  such  parts  that  their  product  shall 
be  a  maximum. 

Hint.     If  x  is  one  part,  then  a  —  xis  the  other,  and  the  function  to 
be  examined  is  r(a  —  x)  or  ax  —  x^.  Equal  parts.  Ans. 

3.  Divide  the  number  a  into  two  such  parts  that  the  product  of  the 
mth  power  of  one  and  the  nth  power  of  the  other  shall  be  a  maximum. 

In  the  ratio  m  :  n.  Ans, 


24.  The  subject  of  Maxima  and  Minima  is  one  of  the 
most  important  in  the  applications  of  the  Calculus  to  Ge- 
ometry, Mechanics,  etc.  It  is  often  necessary  to  derive 
the  expression  for  the  function  to  be  investigated,  and  in 
testing  this,  attention  should  be  paid  to  the  remark  in 
Example  i(/)  of  the  preceding  exercise. 

EXERCISE  13 

1.  A  box  with  a  square  base  and  open  top  is  to  be  constructed  to 
contain  io8  cubic  inches.  What  must  be  its  dimensions  to  require  the 
least  material  ?*  Base  6  inches  square,  height  3  inches.  Ans. 

2.  The  strength  of  a  rectangular  beam  varies  as  the  product  of  the 
breadth  b  and  the  square  of  the  depth  d.  What  are  the  dimensions  of 
the  strongest  beam  that  can  be  cut  from  a  log  whose  cross  section  is  a 
circle  a  inches  in  diameter  ?  f  Breadth  is  J  ^  v/3  inches.  Ans, 


/(^)  =  V^./'(^)=-l-^.  and /''(:.)=- -i-f^  +  -i-f^.  so  that /'(^) 

vanishes  only  if  —  =  o,  and  then/"  {x)  has  the  same  sign  as  — ^. 
dx  dx'^ 

*  Hint.  Let  x  be  the  side  of  the  base,  y  the  height,  then  x'^y  =  108,  i.e,  y  =  —  • 
and  since  the  material  is  x^-]-^  xy,  we  find  by  substituting  for  y  the  function 

X 

t  Hint.  The  strength  therefore  equals  5d'^  multiplied  by  some  constant,  which 
may  be  dropped  by  the  remark  of  §  24.  But  d^  =  a'^  —  d^;  hence  the  function  is 
3(a2— 32),  d  being  the  variable. 


APPLICATIONS 


59 


3.  Find  the  dimensions  of  the  stiffest  beam  that  can  be  cut  from 
the  same  log  as  in  2,  given  that  the  stiffness  varies  as  the  product  of 
the  breadth  and  the  cube  of  the  depth.  Breadth  J  a  inches.  Ans, 


FiG.  15 
4.   The  equation  of  the  path  of  a  projectile  (see  Fig.  16)  is 

y  =  ;irtan  a f ^— , 

2  Vf^^  cos^  a 

where  a  is  the  angle  of  elevation  and  v^  the  initial  velocity.     Find  the 

greatest  height.  V^in2_a^    j^^ 


Y 

<.y 

0 

A^' 

H               ^ 

\ 

/ 

\    ^ 

Fir,.  16 

5.  Find  the  dimensions  of  the  rectangle  of  greatest  area  that  can  be 
inscribed  in  the  ellipse  b'^x'^-\-a^'^=  aP-U^.    Ans.  Sides  are  « V2  and  b\f7.. 

6.  Find  the  altitude  of  the  right  cylinder  of  greatest  volume  inscribed 

Altitude  =  — .    Ans. 
^3 


in  a  sphere  of  radius  r. 


6o  APPLICATIONS 

7.  Assuming  that  the  brightness  of  the  illumination  of  a  surface 
varies  directly  as  the  sine  of  the  angle  under  which  the  light  strikes  the 
surface  and  inversely  as  the 
square  of  the  distance  from  the 
source  of  light,  find  the  height 
of  a  light  placed  directly  over  a; 

the  center  of  a  circle  of  radius 
a  when  the  illumination  of  the 
circumference  is  greatest. 

From   Fig.    17,   the    bright- 
ness at  P  is  given  by 

K  sin  B  _KX  _  X  _     /        x'"-        V^ 

Hence  the  bris^htness  is  a  maximum  when  is  a  maximum. 

^  {a^-^x^y^ 

x—-^.  Alls. 

V2 

25.    Expansion  of  Functions.     By  actual  division 

(19)  -^— =  I  ^x-\-x^  +  ...  +^r^  +  f-^V""''^ 
^          \  —  X  \\  —  x) 

where  n  is  some  positive  integer.     In  this  simple  way  we 

may  find  for  the  function  an  equivalent  polynomial 

all  of  whose  coefficieitts  save  that  of  x^^^  are  constants.     By 
transposition  (19)  becomes 

(20)  — (i  +.:r  +  ;ir2  +  ;r3+  ...  +;r^)  =  — i— .r^+i. 

^  \  —  X        ^  ^         I   —  X 

Now  let  X  be  some  number  numerically  less  than  i,  say 

.r=.5,  and  suppose  we  wish  the  value  of   correct 

within  one  one-hundredth,  i.e,  correct  to  two  decimal  places. 
Let  us  then  determine  for  what  values  of  n   the   term 

-:r^+^  when  x=x  is  less  than  .01,  i.e.  solve  the  in- 

\-x 


equality      __      .5'^"^^  <  .01.     We  find  n  >6. 


APPLICATIONS  6l 

Furthermore,  if  x  is   numerically   less  than  .5,    — 

I  —  X 
and  x''""^  are  less  than  for   ^  =  .5,   so  that  taking    ;/  =  7 

{{.€,  >6),  — — — ;tr^<.oi  for  every  value  of  ;i' not  numeri- 
cally greater  than  .5.  And  we  now  see  from  (20)  that 
the  function       _       may  be  replaced  by  the  polynomial 

I  +  .r  4-  x^  4-  x^  -f-  ;r*  +  ^  +  x"^  +  x"^  for  all  values  of  x 
numerically  equal  to  or  less  than  .5  if  results  correct  only 
to  hundredths'  place  are  desired. 

Precisely  the  same  reasoning  holds  for  any  value  of  x 
numerically  less  than  unity,  since  for  any  such  value  x'^'^^ 
can  be  made  as  small  as  we  please  by  taking  n  sufficiently 
great.  But  this  reasoning  does  not  hold  for  any  value 
equal  to  or  exceeding  i  numerically.  We  may  then  state 
this  theorem  : 

For  any  value  of  x  numerically  less  than  unity ^  the  func- 
tion    7nay  be  represented  zvith  a?ty  desired  degree  of 

accuracy  by  a  sufficiently  great  number  of  terms  of  the 
polynomial^ 

I  +x  -i-x^  -i-x^-h  •••. 

The  Differential  Calculus  enables  us  to  obtain  a  similar 
theorem  for  many  other  functions,  as  will  now  be  explained. 
In  all  practical  computations  results  correct  to  a  certain 
number  of  decimal  places  are  sought,  and  since  the  process 
in  question  replaces  a  function  perhaps  difficult  to  calcu- 
late by  a  polynomial  with  constant  coefficients,  it  is  there- 
fore of  great  practical  importance  in  simplifying  such 
computations. 


62 


APPLICATIONS 


26.  Theorem  of  the  Mean.  If  fix)  and  f{x)  are  con- 
tinuous as  X  varies  from  a  to  b,  then  there  is  at  least  one 
value  of  x^  say  x^,  between  a  and  ^,  such  that 


(21) 


Ab)-f{a)       ,,. 
b-a      -J  ^^1^- 


Fig.  i8 


In  Fig.  1 1,    f{b)  -f{a)  =  CB,   b-a  =  AC, 

/•/ i\  •f(n\ 

/,    /_L_z — ZA-J=  slope  of  AB,  and  at  eack  of  the  points 

P^  and  P^  the  tangent  is  parallel  to  AB^  and  hence  (21)  is 
true  if  x^  is  the  abscissa  of  P-^  or  P^* 
Multiplying  (21)  out  gives 


(22) 


/{b)=f(a)  +  (d-a)/'(x,), 


where  it  must  be  remembered   a>x-^>  d. 

A  more  general  theorem  than  (21)  is  enunciated  as 
follows : 

If  f(x)  and  the  (n -{-  i)  successive  derivatives  /X^)> 
f\x\  •••,  /^"■^^\^)  are  continuous  when  x  varies  from  a 


*  This  proof  of  the  theorem  of  the  mean  is  not  mathematically  rigorous,  but 
merely  illuminates  the  significance  of  (21).  The  student  should  draw  other  figures, 
and  especially  such  that  the  necessary  conditions  of  the  continuity  of  fix)  and 
f\x)  fail. 


APPLICATIONS  63 

to  ^,  then  there  is  at  least  one  value  of  ;r,  say  x^,  between 
a  and  d  such  that 

(23)        Ad)  =f{a)  +  ^^^f{a)  +  ^1:1^  fXa) 

The  proof  of  (23)  is  beyond  the  scope  of  this  book.* 
The  student  should,  however,  carefully  note  the  law  by 
which  the  expression  on  the  right  is  constructed. 

Putting  for  b  in  (23)  the  variable  ;r,  we  get  Taylof^s 
Theorem, 

(24)    /w=/(«)+^-^f^V(^)+^-^j^/"(«)+ - 

yx     a)      y(n+i)(^)  where  a<x.<x. 

Finally,  setting  /2  =  o  in  (24),  we  find  Maclauren's 
Theorem  y 

(25)       /(■«^)=/(o)  +  f /'(o)+^/"(o)+  ...  +^/»>(o) 

+ , — ; — P^'^^x^  where,  o  <  jr.  <  ;r. 
\n  4-  I 

If  in  (23)  we  put  ^  =  ^  +  ;r,  we  obtain  another  form  of  Taylor's 
theorem, 

f{a  +  X)  =f{a)  +  \fia)  +  |V''(^)  +  -  etc. 

This  formula  (25)  gives/(;r)  in  the  form  of  a  polynomial 
in  X  with  constant  coefficients  save  that  of  x^'^^,  which, 
since  x^  lies  between  o  and  x,  is  a  function  of  x ;  that  is, 

*  An  excellent  discussion  is  given  in  Gibson's  An  Elementary  Treatise  on  the 
Calculus,  London,  1901,  p.  390. 


54  APPLICATIONS 

we  have  the  generalization  of   the  example  of   §  25   as - 
follows : 

A  fimctio7i  f(x)  for  certain  values  of  the  variable  *  may 
be  represented  with  any  desired  degree  of  accuracy  by  the 
polynomial^ 

/(o)+^/'(o)  +|/"(o)  +g/'"(o)+  -  +^/<"'(o). 

By  "  expansion  of  a  function "  is  meant  the  forming  of 
this  polynomial.  Of  course  n  is  indefinite,  and  must  be 
taken  great  enough  to  give  the  desired  degree  of  accuracy. 
It  is  of  greatest  theoretical  importance  to  determine  for 
what  values  of  x  the  polynomial  represents  the  function 
when  71  is  taken  indefinitely  great.  This  consists  in  exam- 
ining for  what  values  of  x 

for  this  term  is  the  dijference  between  the  function  and  the 
polynomial. 

EXERCISE   14 

1.  Expand  sin;r. 

Since  f{pc)  =  sin  x,    and  for  x  =  o,    /(o)  =  sin  o  =  o ; 

then  f'(^)  =  cos  x^    and  for  x=  o^  /'(o)  =  i  ; 

/"(;ir)  =  -sin;r,  /"(o)ir:o; 

f"(x)  =  -  cos  X,  f"(o)  =  -  I ; 

/iv(:r)  =  sin  X,  f^(P)  -  o ; 

etc.  etc. 

jy.3  -v-S  -k"  jJ;*9 

Hence  sin  x=  x  — . h , , h , etc. 

[3.     Ll     l_Z.     19.     . 

2.  Show  that  the  expansion  of  cos  x  is 

y-1  •y-4  -y-6  j/^ 

cos  ;r  =  I  - , — 1- , r?:  -^KQ-  ^^c. 

[2      [4      1 6_     II 

*  Namely,  for  all  values  of  x  such  that  the  "remainder"  | — qj-/^**+iH^i)  is 
less  than  the  limit  of  error.    This  question  is  often  difficult  to  settle. 


APPLICATIONS  65 

3.  Expand  eF. 

Since  f[x)=  ^,  and  all  its  derivatives  are  likewise  ^',  while  ^  =  i, 
we  obtain  ^,     ^,     ^     ^., 

Putting  ;r  =  I,  we  find 

I       I       I       I 

the  expression  given  in  §  10. 

The  expansions  of  sin  Xj  cos  x,  and  e'  are  remarkable  in  that  they 
hold  for  eve?y  value  of  Xy  positive  and  negative. 

4.  Prove  the  following  expansions  : 

.   X   ,         ,  XI  fx      x^      x^      x^      x^  \ 

(a)  loga(i  +^)=  ^ogae{^---  +  ~-j-{-j-"'y 

,,s    X  X  m(m  —  i)    ^      ;;/(;//  —  i)(m  —  2)    _ 

(<^)   (I  +  ;r)'«  =  I  +  ///;r+  —^^ -^^  +  — ^^ r^ -^^  +  •••• 

(d)  is  the  binomial  formula.  These  expansions  hold  only  for  values 
of  x  numerically  less  than  i . 

Taylors  Theorem  (24)  differs  from  (25)  in  that  we  are 
to  consider  values  of  the  variable  x  near  some  given  num- 
ber Uy  since  (24)  is  a  polynomial  in  (x  —  a)  in  the  same 
sense  that  (25)  is  a  polynomial  in  x.  It  is  evident  that  no 
greater  difficulty  arises  in  the  application  of  (24)  to  a  given 
function  than  has  been  already  pointed  out. 

27.  Differentials.  From  (23)  we  are  able  to  find  an  ex- 
pansion for  the  increment  of  a  function  in  powers  of  the 
increment  of  the  variable  as  follows  : 

Write  b  =  X'\-/^Xy  a  =  Xy  .*.  b  —  a  =  Ax,  and  (23)  be- 
comes, after  transposing  /(x)y 

(26)     /(x  +  A;r)  -/(x)  =  Axf(x)  +  ^/"(^)  +  -, 

or  (27)        A/ix)  =/'(x)Ax  +/"(x) ^-^  +  .... 

Now,  if  we  suppose  Ax  to  diminish  toward  zero,  the  first 
term  /'{x)Ax  of  the  right-hand  member  will  ultimately 

EL.  CALC.  —  5 


66  APPLICATIONS 

greatly  exceed  the  sum  of  the  remaining  terms,  since  these 
contain  higher  powers  of  A;r.  For  this  reason  /'(x)Ax  is 
called  the  principal  part  of  the  increment  of  fix).  Also, 
when  we  wish  to  emphasize  the  fact  that  the  variable  l^x 
is  to  approach  zero  as  a  limit,  we  write  dx,  called  differen- 
tial X,  instead  of  A;ir,  and  the  principal  part  of  the  incre- 
ment/'(;r)^^  we  call  the  differential  of  the  function  ;  that  is, 
(28)  df{x)=f'{x)dx. 

The  following  definitions  are  fundamental: 

A  differential  (or  infinitesimal)  is  a  variable  whose  limit 
is  zero. 

The  differential  of  the  independent  variable  is  an  incre- 
ment of  that  variable  whose  limit  is  zero. 

The  differential  of  the  dependent  variable  is  the  princi- 
pal part  of  the  increment  of  that  variable,  and  equals  the 
product  of  the  derivative  a7td  the  differential  of  the  inde- 
pendent variable  (28). 

From  (28),  we  see  that  if  7  is  a  function  of  x,  then 

EXERCISE   15 
1.   Prove  by  (28)  and  (29)  the  following  differentials : 
{a)   d{2,x'')=6xdx,  (^)    dy/V^x  = ^ 


dx  2W^r^ 

{b)    d\og,x  =  —'  (f)  dsm2x  =  2cos2xdx, 

(c)  de'  =  e^dx.  sec^f-) 

(d)  dx^  =  mx-^-^dx.  (^)   ^  t^^  (^)  = jr- ^^' 

(Ji)    \i  y  ■=  xlogeX,  then  dy  =(i  +  loge:r)  dx. 
2.    U  y  =  uvy  then 

dy  =  (u^ -{-  v"^)  dx  =  u~  dx  +  V  —  dx,  or  dy  =  udv  +  vdu. 
\    dx         dx)  dx  dx 


3.    Show  that 


APPLICATIONS 
V  du  —  u  dv 


67 


'(j)^ 


4.  State  the  rules  I-V  for  differentiation  in  terms  of  differentials 
instead  of  derivatives. 

28.    We  may  write  (27)  after  replacing  A;r  by  dx, 
(30)     ^f(^x)^f\xyx^dx''{^^^^^^ 

Now,  since  by  (28)  f{x)dx  is  the  differential  of  the 
function,  (30)  shows  that  A/(,r)  and  df{x^  differ  by  a  term 
containing  the  factor  dx'^.  Such  a  quantity  is  called  a 
differential  of  the  second  order ;  in  general,  any  quantity 
containing  as  a  factor  the  product  of  tivo  differentials  is 
thus  designated. 

The  increment  of  a  function  differs  from  its  differential 
by  a  differential  of  the  second  order. 

EXAMPLES 
1.   Differential  of  a  product  nv, 

0'   i 


^ 


udv 


^  m^' 


E' 


du 


Fig.  19 


B    B' 


Let   71  =  AB,  V  =  ACy    then    uv  =  area    ABCD.      If   du  =  BB\ 
dv  —  CC\  then 

t^{t(v)  =  area  AB'CD'  -  area  ABCD 

=  area  CDCE  +  area  BB'DE'  +  area  DE'ED' 
=  udv  -\-  V du  +  du  •  dv. 
Now  du  '  dv  is  a  differential  of  the  second  order,  .-.  principal  part 
oi  ti(uv)  \s  udv  +  vdu\  i.e.  d{uv)=  udv  ■\-  vdu.     (Cf.  Ex.  2,  §  27.) 


68  APPLICATIONS 

2 .    Differential  of  an  area . 


X' 


dx 


N 


Fig.  20 


X 


Consider  the  area  aAPM  bounded  by  any  curve,  the  axis  XX'  and 
the  ordinates  aA^  MP,  and  call  this  area  ?/.  Then  if  MN=  dx,  t^u 
=  area  a  A  QN-  area  aAPM-  area  MPQN.  r .  ^u  -  y  dx -\-  area  PSQ . 
But  area  PSQ  <dx'  dy,  .  • .  PSQ  =  k  dxdy,  where  k  is  some  number  <  i . 
Hence  area  PSQ  is  a  differential  of  the  second  order,  and  .*.  du=y dx. 

The  differ eiitial  of  the  area  bounded  by  any  curve,  the  axis  XX' ,  and 
two  ordinates  is  the  product  of  the  ordinate  of  the  curve  and  the  differ- 
ential of  the  abscissa. 

3.   Differential  of  the  volume  of  a  solid  of  revolution. 

Let  the  solid  be  generated 
by  revolving  a  curve  APQ 
around  XX',  and  denote  the 
volume  APA'P  by  v.  If 
dx  —  MN,  then  At/  =  volume 
AQA'Q'  -  volume  APA'P', 
or  At/  =  volume  of  the  cylin- 
der PSP'S'  H-  volume  gener- 
ated by  the  curvilinear  A  PSQ. 
Now  the  volume  of  the  cylin- 
der PSP'S'  —  iry-dx,  since  / 
=  PM  =  radius  of  base  and 
^jr=  altitude.  The  volume 
generated  by  the  curvilinear 
A  PSQ  <  volume  generated 
by  the  rectangle  PRSQ,  and 
this  last  volume  =  ttA^^  •  MN  -  ttMP^  .  MN z=:  it  {2 y  dy  ^  dy^)dx. 


Fig.  21 


APPLICATIONS 


69 


We  see  therefore  that  At/ =  tt/^  ^/r -f  a  differential  of  the  second 
order,  i.e.  dv  —  iry'^dx. 

The  differential  of  the  volume  of  a  solid  of  revolution  generated  by 
revolving  any  curve  around  the  axis  XX '  equals  ir  times  the  product  of 
the  square  of  the  ordinate  and  the  differential  of  the  abscissa. 


Q{r+dr,d+dd) 


P{r,e) 


Fig.  22 


4.  Show  that  the  differential  of  the  area  u  bounded  by  a  cur\'e  AP 
and  two  radii  vectores  OA  and  OP  is  given  by  du  =  ^r^dO,  where 
(r,  0)  are  the  polar  coordinates  of  P. 


CHAPTER   IV 
INTEGRATION 

29.   Indefinite  Integral.     Integration  consists  in  finding 
a  function  of  which  a  given  differential  expression,  such  as 

dzi 
X  dx^  sinxdxy  — ,  etc.,  is  the  differential.     The  function 

thus  found  is  called  the  integral  of  the  given  differential 
expression,  and  the  operation  is  indicated  by  prefixing  the 

integral  sign   j  .     Thus,  since 

d{^  x^)  =  X  dx,       .  • .    \  xdx=  ^x^; 

I  dx  =  Xy         j  sinxdx=  —  cos  x,  etc. 
In  general, 


// 


means  to  find  a  function  F(x)  such  that 
dF(x)  =f{x)dx. 

Constant  of  Integration.     Since  d(^  x'^  4-  C)  also  equals 
X  dxy  no  matter  what  the  constant  C  is,  we  have 


\xdx  =  ^x^  +  Cy 


where  C  is  any  constant  whatever,  called  the  constant  of 
integration.     We  see,  therefore,  that  a  given  differential 

70 


INTEGRATION  Jl 

expression  may  have  infinitely  many  integrals,  found  by 
giving  to  the  constant  of  integration  different  values. 
Thus 

and  since  C  is  unknown  and  mdefinite^  ^{^)  -h  C  is  called 
the  indefinite  integral  of  f(x)dx. 

Of  course,  the  same  differential  expression  has  an  in- 
definite number  of  distiiict  integrals,  but  what  has  just 
been  said  shows  that  the  difference  of  any  two  of  these 
must  be  a  constant. 

30.  Rules  for  Integration.  From  Rule  V  in  differentia- 
tion, if  V  is  any  function  of  x,  and  ic  a  constant,  then 

^-.(icv)  =  /c— ,     i.e.  d(Kv)  =  fcdv. 
dx  dx 

Integrating,  we  have,  since  if  two  differential  expressions 
are  equal  so  are  their  integrals  equal, 

j  fcdv  =  1  d{f€v), 

or,  since  j  d(Kv)  =  kv, 

Kv  =  j  f€  dv. 

But  K  j  dv  =  fcv. 

(31)  .',   j  icdv  =  fc  j  dv. 

XXIII.  A  constant  factor  may  be  written  either  before  or  after 
tlie  integral  sign. 

The  chief  application  of  XXIII  is  to  be  found  in  cases  like  the 
following : 


72  INTEGRATION 

To  work  out  i  xdx.     If  we  multiply  xdx  by  2,  we  have  an  exact 

differential,  since 

d  (;i'2)  =  2  xdx, 

.*.    \2xdx  =  x^  ; 

but  by  XXIII,  j  2  ;ir^ar  =  2  i  ;if ^;»r, 

.-.    (xdx=^. 
J  2 

From  (31)  we  may  also  write 

(32)  ff(x)  dx=^  ffc/(x)  dx. 

Integral  of  a  Sum  of  Differential  Expressions.     If  u  and  v 
are  functions  of  x,  then 

diu  +  z/)  =  —  {ti  '\'  v)  dx  —  du  -{■  dv. 
dx 


*.    j  (<^;^  +  dv)  =  j  <3f(^  +  Zf)^  u  -\-  V  =  \  dii-\-  \  c 


dv. 


This  result  gives  Rule 


XXIY.  The  integral  of  any  algebraic  sum  of  differential  ex- 
pressions eqnals  the  same  algebraic  sum  of  the  integrals  of  these 
expressions  taken  separately. 

That  is,  e.g., 

J  (-^  +  3)  dx=  \  {xdx  +  3 dx)  =  \xdx  +  \;^dx=\x'^-\-^x-\-  C 

31.  From  any  result  in  differentiation  may  always  be 
derived  an  integration  formula,  and  we  now  proceed  to 
obtain  some  of  the  simpler  ones,  making  use  of  §  18. 


INTEGRATION  73 

Since  by  VIII, 

then,  integrating, 

^m+i^  r(w-h  i)v'^ dv  =^  {m  ^-  \)^v^dv,    (XXIII) 

From  IX,  ^log,v=— , 

rdv      , 
(34)  •*•  JV  =  ^^^^''- 

In  the  same  way  we  might  go  through  with  each  formula 
in  §  1 8.  It  will  suffice  for  our  purpose  to. tabulate  a  few 
of  the  results : 

XXV.   fi;»^cfv  =  ^^^^  +  C(m:?^-l). 
XXVI.   f^  =  logei^+C. 

J    V 

XXVII.    fat^<ft;  =  r-^^  +  C. 
J  loge  a 

XXVIII.  J  sin  vdv  =  -  cos  v  +  C. 
XXIX.    f cos  V  cfv  =  sin  V  +  C 
XXX.   f       ^^       =:arcsin^+C. 

XXXI.    f_^  =  ?  arctan^  +  C. 


74 


1.   Find 


INTEGRATION 

EXAMPLES 

dx 


C 


X 


This  is  the  same  thing  as  f  (i  —  x)  ^ dx^  which  resembles  XXV. 

For  put 

I  —  x-=v^  then   —  dx  :=  dv,  or  dx  —  —  dv. 

.'.    \i}  —  x)~^dx  =  \v~^  —  dv  —  —  \v~^dv. 

.-.  by  XXV  {v'^dv^'^^C, 

and  by  substituting  again, 

•^  Vl  —  ;r 

2.   Workout  C3axdx_ 

Jc^-  d^x^ 

Taking  out  constant  factor  3  a  (XXIII),  this  becomes 

r     xdx 

^"^J  c^-  d^x^' 
and  this  resembles  XXVI. 

For  put  6-2  —  dH-^  =  Vf  .'.   —  2  b'^xdx  =  dv,  or  xdx  = ^. 

2  a^ 

_  -^ 
r     xdx  r     2d^         7.  a  C dv         3^1  ,^ 


3.   Find 


/; 


^;r 


9  +  4^' 

This  resembles  XXXI,  if  ^  =  3,  ix=v. 

Then  2  dx  —  dv,  and  since  the  given  integral  by  (32)  is  the  same  as 
If       2dx  I  C     dv 


f       "^dx  £  C 

2}  32+(2;r)2  ^^  2  J  ^2_,_^2' 

we  find  by  XXXI,      f      ^^    „  =  ^  arc  tan  —  +  C. 
^  9  +  4^2     6  3 


INTEGRATION  75 

By  studying  the  above  examples  the  student  will  see 
that  integration  depends  upon  comparison  of  the  given 
integral  with  certain  standard  forms.  To  be  able  to  tell 
quickly  what. form  the  given  integral  resembles  is  abso- 
lutely essential. 

Tables  of  standard  forms  *  have  been  constructed  con- 
taining all  integrals  occurring  in  ordinary  work. 

EXERCISE   16 

1.  Prove  the  following  integrations  : 

(a)    J  {ax  +  dx^)  dx=lax'^-\-\  bx^  +  C.     (Use  XXIV.) 

(jf)    1  =  loge (I  -  COS  x^  4-  C. 

J    I   -  COS  Jf 

(0  r  V^2  _  x'lxdx  =  -  \{a?'  -  x'^Y  +  C'  (UseXXV,  2/  =  fl2_;^a'). 

{d)  \  sin  {7.x)dx=i  —  J  cos  2  ;ir  +  C 

(.)  ^e-^dx  =->r-'+C.                      ig)  J  ^^  =  V55+^+  C. 

(/)  f— ^=:  =  iarcsin(2x)  +  C.  W  CJ^  =  -\og,{i-x)  +  C. 

•^    VI  A  x^        2  •/  I  —  X 

(0     U-xdx=ixUC;     r^=--L+c. 

^  ^  X^  2X-^ 

C  sin  X 

(J)   \t2inxdx=  -  loge  COS  X  -\-  C.      (Put     tan  x  = and    use 

•^  cos  X 

XXVI.) 

(k)    isiu'^xdx  =  ix -  isin2x-\- C.     (Put  sin^ji- =  ^(i  -  cos  2r).) 

2.  Special  Devices  in  Integration. 

(a)  By  partial  fractions^  when  we  have  to  integrate  a  rational  frac- 
tion times  dx^  and  this  fraction  can  be  replaced  by  partial  fractions. 

•  E.g.,  B.  O.  Peirce's  A  Short  Table  of  Integrals,  Ginn  &  Co..  1899. 


76 

INTEGRATION 

For  example, 

Putting 

1       _     A            B 

a^ 

-x^      a  -X     a-\-  X 

and  clearing  of  fractions, 

I  = 

x(A-B)-{-a(A+B), 

.-.  A- 

-  B  =  o, 

,  a(A  +B)=  I,   or  A 

I 
'2a 


Ja^-x^     2aJ  a  -  X     2aJ  a-\-  X     ^a>    "  *■         '       "'^         " 

2a     ^  \a  -  xj 
(d)   By  change  of  variable. 
Find    I  \/^2  _  ^1  ^x.     Substitute  ;ir  =  ^  cos  ^ ; 


.-.  dx  =  -a?:mBdO,  Va^  -  x^  =\/a^  -  a^cos'^O  =  asinO, 
and  J\/«2  _  ^2^^  ^  _  a'^^sm'^Odd  =  -  ^  ^  +  ^  sin  2  ^  +  C 
by  Ex.  I  (y^).     Now 

X  I  X^      X 

0  =  arc  cos-,   sin  2^  =  2  sin ^ cos ^  =  2\/i 5  •  -. 

a  \        a^    a 

.'.    \  V^2  _  ^2  ^x  = arc  cos  -  +  -  xVa"^  —  x^. 

J  2  a      2 


3.  Prove  f.^— ^  log,^fZI+ c. 

The  following  two  examples  illustrate  the  manner  of  determination  of 
the  constant  of  integration  by  means  of  so-called  initial  conditions. 

4.  Find  the  amount  of  a  sum  of  money  increasing  continuously  at 
compound  interest  of  r  per  cent. 

We  found,  page  43,  that,  in  derivatives,  P  being  the  sum  sought, 

dP^jr_p 
dt      100 
Multiplying  by  dt  and  dividing  by  /*,  we  have 

P       100      ' 
integrating,  ( i )     log,  Z'  =  ^  /  +  C 


INTEGRATION 


77 


Let  now  a  equal  the  iniiial  sum  of  money ;  that  is,  the  sum  started 
with,  so  that  P  =  a  when  /  =  o ;  substituting  these  in  this  equation, 

we  have  log^^  =  C,  so  that   (i)   becomes   loge/'  =  —  /  +  loge«,  or, 

transposing,  ^  i  P\       r 

loga/^-log.^  =  — /,   or   loge^-j  =  — /; 


lOO 


Ans. 


5.  Find  the  relation  between  s  (space)  and  t  (time)  for  uniformly 
accelerated  rectilinear  motion. 

Since  the  acceleration  — ^  is  constant,  say  /.  we  have  —  —  f. 

Multiplying  by  dty  dv  —fdt,  and  integrating,  v  =ft  +  C. 
To  determine  C,  let  the  iyiitial  velocity  be  v^^  i.e.  v  =  Vq  when  t  =  o, 
or  z/q  =  o  +  C.     .'.  V  =ft  +  Vq. 

Since  v  =  —,  .-.  -^  —ft-\-v^^  and  multiplying  by  dt^  ds—ftdt-\-Vrdt. 

dt         dt 
Integrating,  s  =  \ft'^  +  vj-  +  C,  and  if  s  —  s^  when  /  =  o,  we  have 
finally  s  =  J//-^  +  v^t  +  Sq.     Ans. 

32.  Definite  Integral.  We  have  already  seen  that  the 
indefinite  integral  contains  an  arbitrary  constant,  t/ie  co?z- 
stant  of  integration,  and  has  for  that  reason  an  indefinite 
value.  By  making  suitable  assumptions,  now  to  be  ex- 
plained, we  are  able  to  dispose  of  this  inconvenience. 

In  §  28,  Example  2,  it  was  shown  that  the  differential  of 

the   area   ti    between   a 


curve  MABCy  the   axis 
XX' y  and  any  two  ordi- 
nates  was  given  by 
du  =y  dx. 

.'.  u=  \y  dx ■\-  C. 

Here,  of  course,  y  is 
some  function  of  x  determined  from  the  equation  of  the 

curve,  and  .*.    \ydx-=-  some  function  of  x,  say  F{x\ 

.-.  //=F(;r)-hC 


Fig.  23 


78 


INTEGRATION 


Let  US  now  agree  to  reckon  the  area  from  the  axis  VV^, 
so  that  when  x  —  a^  tc=^  area  OaAM,  etc. 

Under  this  assumption,  when  ;r  =  o,  u  =  o,  and 

.-.  o=i^(o)+C,    or    C^-Fip), 

and  we  have 

u  =  F{x)  —  F(o). 

Now    area  OaAM  =  F{a)  -  F{o). 

Area  ObBM  =  F{b)  —  F(o).    Subtracting,  we  have 

Are^  aMB  ^F{b)-F{a), 
or. 

The  difference  of  values  of  the   \  y  dx  for  x  —  b  and  x  =  a 

gives  the  area  bounded  by  the  curve  zvhose  ordinate  is  y,  the 
axis  XX' y  and  the  ordinate s  at  a  and  b. 

This  difference  is  represented  by  the  symbol 

(35)  jydx, 

read,  "  integral  from  a  to  b  oi  y  dx''  ]  the  operation  is  called 
integration  between  limits ^  a  being  the  lower ,  b  the  upper 
limit. 

We  see  therefore  that  (35)  or,  what  is  the  same  thing, 

(36)  £A^yx 

always  has  a  definite  value,  and  is  accordingly  a  definite 
integral.     For  if 

(37)  ^f{x)dx  =  F{x)-^C,  then 

(38)  £f{x)dx^  F{b)  +  C-  {F{a)  +  C)  =  F(d)  -  F{a), 
and  the  constant  of  integratioft  has  disappeared. 


INTEGRATION  79 

33.   Areas  of  Plane  Curves.     From   §  32,  we  have  the 
theorem :    Given   any  plane  curve  y  =zf(x\    the   definite 

f{x)dx  gives  the  area  bounded  by  that  curve ^ 

the  axis  XX'  and  the  ordinates  at  a  and  b. 

To  find  the  area  bounded  by  two  given  curves,  we  get 
the  area  between  each  and  XX^  and  then  subtract. 

Volumes  of  solids  of  revolution. 

Precisely  as  in    §   32  and   remembering  the  result  of 
Example  3,  §  28  we  prove  that : 

Given   any  plane  curve  y  =f{x)y    the   definite   integral 

1    iry'^dx  gives  the  volume  generated  by  revolving  around 

XX'  the  portion  of  the  curve  between  the  or  dictates  at  a 
and  b. 

The  two  theorems  just  given  find  numerous  applications 
in  Geometry. 

EXERCISE   17 

1.  Find  the  area  of  the  curve  /  =  ;ir2  —  9  lying  below  XX' . 

Here    \  ydx=  \  (x^-  9)  dx,  and  since  for  /  =  o,  ;r  =  ±  3,  the  limits 
are  +3  and  —3,  z.e.  area  =  i     (x^—g)dx.  36.  Ans. 

2.  Find  the  area  of  the  circle  x^-\-y'^  =  d^. 

Since  y  —  ^a^  —^%  \ydx=\  y/a^—  x^dx  which  has  been  worked 

out  in  Exercise   16,  Example  2  {U).      For  the  semicircle  the  limits 
are  4-  a  and  —  a. 

3.  Show  that  the  area  of  the  ellipse  b'^x'^-\-  d'-y'^  =  d'-b'^  is  to  the  area 
of  the  circle  whose  diameter  is  the  major  axis  2a2iS>b\a, 

4.  Find  the  area  of  one  arch  of  sine  curve  y  =  sin  x.  2.  Ans. 

5.  Find   area  between  the  equilateral  hyperbola  xy=i.  the  axis 
XX' J  and  the  ordinates  at  x=ay  x  =  3.  loge  ( -  )  •  Ans. 


So 


INTEGRATION 


6.  Find  the  volume  of  the  sphere. 

Since  we  have  to  revolve  the  circle  x^-\-y^  =  a%  or  y^  =  a^—x^ 
around  XX',  then  J  iry^^x  =  ttJ  (a^  -x'^)  dx.  The  limits  are  +  a  and 
-a.  ^ira^.  Ans, 

7.  Find  the  volume  generated  by  revolving  around  XX'  the  pa- 
rabola ^^^  —  ^  ^^  and  cut  off  by  a  plane  perpendicular  to  XX'  at  the 
distance  of  4  to  the  right  of  the  origin.  32.  Ans. 

34.  Definite  Integral  as  the  Limit  of  a  Sum  of  Differential 
Expressions.  In  the  Differential  Calculus  the  student  was 
asked  to  bear  in  mind  that  everything  was  built  up  from  a 
fundamental  limit,  the  limit  of  a  quotient  whose  denominator 
approached  zero.  We  are  now  to  see  that  the  definite  inte- 
gral is  the  limit  of  a  sum  of  differential  expressions. 


If 


^f{x)dx^F{x)^C, 


then     4^W=/Wand    C f(x)dx  =-F{b)-F{a) 
dx  ^^ 

gives  the  area  bounded  by  the  curve7=/(;r)  (Fig.  24), 

the  axis  XX\  and  the  ordinates  ^tx=^a,x=^b. 


< — AjX — > 


Pr— 


^, 


P,r- 


bi     X2     h^     X3     bs     a?4     64     a?5 
Fig.  24 


Q 


Now  divide  the  segment  ad  into  any  number  of  equal 
parts,  say  6,  a^b^=^b^b^=  •..  =^b^b,  and  call  the  length  of 
each  division  Lx.    Erect  the  ordinates  at  these  points,  and 


INTEGRATION  8 1 

apply  the  theorem  of  the  mean  (§  26)  to  each  division. 
In  the  present  case  F(x)  takes  the  place  oi  /(x)  in  (21), 
and  /{x)  replaces  /'(x) ;  for  the  first  interval  ad^,  a=^a, 
b  =  by  and  x^,  lying  between  a  and  b,  is  marked  in  the  figure. 
Draw  the  ordinate  of  Xy     Then  (21)  gives 

F{b,)-Fia) 

or,  since  b-^  —  a  =  Ax, 

(39)  F{b,) -F{a)=f  {x,)A.r. 

In  the  same  way  (21)  applied  to  each  of  the  remaining 
five  segments  gives  the  equations 

■F{b,)-F{b,)=Ax,)A.r, 
F{b,)-F{b^)=/{x,)Ax, 

(40)  \F{b,)-F(b,)=/{x,)Ax, 

F{b,)-F{b,)=/{x,)Ax, 
F{b)  -Fib,)=/(x,)Ax. 

Adding  the  six  equations  (39)  and  (40),  we  find 

(41)  F{b)-F{a)=/{x,)Ax+f(x,)Ax+/Qr,)Ax 

+/{x^)Ax+/{x,)Ax+/{x,)Ax. 

But     /{xj)Ax  =  area  of  the  rectangle  aPP-^b-^, 
f{x^Ax  =  area  of  the  rectangle  b-^p^P^b.,, 
etc., 
so  that  the  sum  on  the  right  equals  the  area 

aPPxPxP'LpiP%PzP^Pj'f.P^Qb->  i-e- 

(42)  F{b)  —F{a)  =  area  between  the  broken  line 

/'/'lA-AGand^^', 


82  INTEGRATION 

and  this  is  true  independently  of  the  number  of  parts  into 
which  ab  is  divided.  Hence  for  any  number  n  of  equal 
parts 

(43)  F{b)-F{a)  =f{x^^x  -\-f{x^)^x  +  ...  +/(^,)A^, 

(44)  and  A;r  = 

Equations  (43)  and  (44)  hold  when  n  increases  without 
limit,  and  then  A;r  becomes  dx  (§  27),  ie,  a  variable  whose 
limit  is  zero. 

.-.  F{b)-F{a)=  l;'^'^ {f{x^)dx-\-f{x^)dx+^^>-^f{x,)dx\ 
or,  by  (38), 

(45)  £A^y^  =  n'=L{A^iy^+A^^y^+  ...  +/(;r,yr). 

And  now  we  see  very  clearly  why  J^  f{x)dx  gives  the 
area  under  the  curve,  for  as  n  increases,  the  broken  Hne 
PPxP\P^Pi  '"pr^Q  approaches  the  curve  itself,  and  the  sum 

f{x^dx-\ \-f{x^dx  always  represents  the  area  under 

this  broken  line. 

Integrating  between  limits  is  accordingly  spoken  of  as 

"summing  up";   the  integration  sign    I    is  historically  a 

distorted  5,  the  first  letter  of  sum.  But  let  the  student 
not  forget  that  the  definite  integral  is  not  a  sum,  but  the 
limit  of  a  sum,  the  number  of  terms  increasing  ivithout 
limit,  and  each  term  itself  diminishing  toivard  zero. 

The  problem  of  finding  the  area  is  then  to  be  thought 
of  thus :  Divide  the  interval  on  xx^  into  any  number  of 
equal  parts,  and  at  a  point  within  each  division  erect  an 
ordinate  to  the  curve ;  construct  the  rectangles  on  the 
divisions    as   bases,  with   the   corresponding   ordinate   as 


INTEGRATION  83 

altitude.  Then  finding  the  area  consists  in  summing  up 
these  rectangles  and  taking  the  limit  of  this  sum  as  the 
number  of  divisions  increases  without  limit. 

As  an  example  of  the  great  number  of  problems  in  Physics  and  other 
branches  of  Mathematics  which  involve  in  their  solution  definite  inte- 
grals, consider  the  following : 

To  determine  the  amount  of  attraction  exerted  by  a  thin,  straight, 
homogeneous  rod  of  uniform  thickness  and  of  length  /  upon  a  material 
point  P  of  mass  ///,  situated  in  the  line  of  direction  of  the  rod. 

, ^£-H H a >P 


Fig.  25 

Imagine  the  rod  (see  Fig.  25)  divided  up  into  equal  infinitesimal 
portions  (elements)  of  length  dx.     If  M  —  mass  of  rod,  then 

M 

—  dx  =  mass  of  any  element. 

The  law  of  attraction  being  Newton's  Law,  i.e.  attraction  =  product 
of  masses  ^  square  of  distance,  then 


attraction  of  element  dx  on  P  —  ■ 


^mdx 


and  the  total  attraction  is  the  su?n  of  these  from  x  —  o  to  x  -  I. 

—  VI  dx 

...  Force  =  Vl ^^T      "^      ^ 

Jo  {x^aY         I    Jo  {x-\-  ay 

or  integrating,       Force  =^(-  —L-  +  -^  =  -    /^^^^  .    Answer. 


CHAPTER   V 

PARTIAL   DERIVATIVES 

35.  Functions  of  More  than  One  Variable.  In  the  pre- 
ceding chapters  we  have  been  concerned  with  functions  of 
one  variable ;  i.e.  the  variable  function  depended  for  its 
value  upon  the  value  of  a  single  variable.  Such  functions 
do  not  by  any  means  suffice  for  the  applications  of  the 
Calculus.  In  fact,  the  student  is  already  famiHar  with 
many  examples  of  a  variable  whose  value  depends  upon 
those  assigned  to  two  or  more  distinct  variables.  Thus 
the  area  of  a  rectangle  is  a  function  of  two  variables,  viz. 
the  two  sides ;  the  volume  of  a  gas  depends  upon  both  the 
pressure  and  the  temperature ;  the-  volume  of  a  parallele- 
piped depends  upon  the  three  edges,  etc. 

Notation.  If  the  value  of  a  variable  u  depends  upon 
two  variables,  x  and  j/,  and  can  be  computed  when  values 
are  assumed  for  x  and  y,  then  we  write  precisely  as  in  §  3, 

(46)  u-=f{x,y). 

Similarly  for  a  function  of  three  variables, 

u  =  ^(x,y,  z)y  etc. 

36.  Partial  Differentiation.  As  in  §  12  the  important 
question  arising  here  is  how  to  determine  the  manner  of 
variation  of  the  function  when  the  variables  change  in 
value.  But  we  have  greater  latitude  here  than  in  §  12. 
For  in  (46)  we  can  ask  ourselves, 

84 


PARTIAL   DERIVATIVES  85 

first,  how  does  71  vary  when  x  alone  varies  and  y  remains 
constant  ?  or 

second,  how  does  u  change  when  x  remains  constant  and 
y  varies  ?  or 

third,  in  what  manner  does  //  vary  when  both  x  and  y 
change  independently  of  each  other  ? 

Thus  let  «  =  JTK,  X  and  /  being  respectively  the  base  and  altitude 
of  a  rectangle  ;  if  /  remains  constant  (say  /  =  b),  u  gives  the  area  of 
all  rectangles  of  a  certain  altitude  b\  and  if  ;t-  =  a  constant,  say  a,  then 
u  represents  the  area  of  all  rectangles  with  common  base  a.  But  if 
X  and  /  both  vary  independently,  then  we  are  to  consider  all  possible 
rectangles. 

Now  the  first  and  second  cases  do  not  differ  in  the  least 
from  §  12,  for  we  really  have  in  \}i\^  first,  u  a  function  of  x 
alone,  and  in  the  second,  u  a  function  of  y  alone.  We  can 
therefore  form, 

first,  the  increment  quotient  (§  13)  when  x  alone  varies, 
and  this  is 

(47)  ^  =  f{x^Lx,y)-f{x,y) 

^^'  ^  Ax  Ax 

second,  the  increment  quotient  when  y  alone  varies,  which 
*  is 

/^ox                     A?^  _  f{x,  y  +  Ar)  -f{x,  y) 
^^^^  ^  " A^ "• 

For  example,  in  the  area  of  rectangle  already  used,  u  =  xy, 

Au     (x  +  Ax^y  -  xy  ,   A?^       , 

^  = -^ =/,  and  ^-    reduces  to  ^. 

Finally,  we  can,  as  in  §  14,  find  the  limits  of  the  func- 
tions in  the  right-hand  members  of  (47)  and  (48),  in  (47) 
when  Ax  approaches  zero,  in  (48)  when  Ay  approaches 


S6  PARTIAL   DERIVATIVES 

zero.  The  results  are  called  the  partial  derivatives  of  u 
or  f{x,  J/)  with  respect  to  x  and  j/  respectively,  and  this 
step  of  passing  to  the  hmit  we  indicate  on  the  left  by 
replacing  the  A's  by  round  6's,  so  that 

(49)  ^  =  Lmnt  (l^ A.  1^.> 


(so)  — -  =  Limit — \ 

The  partial  derivatives  — ,  —  are  then  to  be  calculated 

dx    dy 

by  the  rules  of  Chapter  II,  the  independent  variables  being 
respectively  x  and  y. 

EXERCISE   18 
1.   Find  the  partial  derivatives  of: 


dx       X    dy    y 


(2)  ^<f  =  arc  tan  f  =^  j . 


Ans.  ^^  = 


d]i_  _     y        bu  _      X 
dx~      x^  +/2'  Sy  ~  x^+y^' 

(3)  u  =  xv.  Ans.  ^  =yx^-^ ;  ^  =  ^^  loge  x, 

dx  ay 

Partial  Differentials : 

By  §  27,  (29),  the  differential  of  ?/,  when  ;r  alone  varies  is 

—  dxy  and  when  y  alone  varies  equals   —  dy\   these  are 
dx  dy 

called  the  partial  differentials  of  u. 

—  dx=  partial  differential  of  ti,  when  x  alone  varies ; 

.     .      dx 

^^'^  ^  d» 

—  dy  =  partial  differential  of  ^/,  when  y  alone  vanes. 
dy 


PARTIAL   DERIVATIVES  Sy 

37.    Total  Differentiation.     We  have  yet  to  discuss  the  third  case 
of  S  36,  viz.  required  the  change  in  u  when  x  and/  vary  independently. 
If  Ao-,  A/,  and  A//  are  the  increments  of  these  variables,  then  from  (46) 
we  have 
(52)  A//  =/(-r  H-  Ar,  /  +  A/)  -/(r,  /). 

By  adding  and  subtracting  /(r,  /  +  A/)  in  the  right-hand  member, 

(52)  becomes 

(53)  ^'^  =/(•*'+  ^-^^  ^^ + ^-^)  ~/(^'  -^  +  '^■^'^  +-^^-''  y'^^y^  ~f^^'  ^^ ' 
Consider  now  the  last  two  terms, 

/{x,y-\-^y)  -/(r,y). 
This  is  the  increment  of  ?/  or  /(x,y)  when  y  alone  varies.     Hence, 
by  (27),  §27, 

(54)  /(^'/  +  ^/)  -/(^'^y)  =  Y  ^^  "^  ^^^^^  ^^  ^^^^^^  powers  of  A/. 
In  the  same  way  the  first  two  terms  of  (53)  give  us,  if  we  set 

u'  =:f(x,y  +  ^y)y 

(55)  /(^+ Ar,/  + Aj)  -/(^,/  + Ay) 

=  ^  A;i'  +  terms  involving  Kx^,  etc. 

But  also 

,/  =/(:., 7  +  A;.)  =Ar,y)  +  |a>^  +  terms  in  ^/,  etc. 

by  (26),  §  27.     Differentiating  with  respect  to  x,  we  find 
.^.  57£:  ^  ^  4.  terms  in  A/, 

since  u=/(r,y). 

Consequently,  from  (56),  (55^,  and  C54),  (53)  becomes 

/  -  ^x     ^u  =  ^'  Ar  4-  ^  A/  +  terms  of  higher  degree  in  Ar,  A/. 
v57;  ^jr  c)y 

Now  letting  A.- and  ^y  approach  zero,  /...  become  ^'^^"fi"';"™  J 
rf^  and  ^/,  then,  as  in  §  27,  calhng  the  prinapai  part  of  A«  the  /./^/ 
differential  of  ?/,  we  have 

(58)  ■'"  =  |.*+|* 


88  PARTIAL   DERIVATIVES 

From  (51)  and  (58),  then,  we  have  the  theorem : 

The  total  differential  of  a  ftmctiofi  of  several  variables 
equals  the  sum  of  the  partial  differentials. 

Example.     In  §  28,  Example  i,  was  demonstrated  the  result 
d  (xy)  =  xdy  +  /  dx^ 


which  agrees  with  (58). 

EXERCISE   19 

Find  the  total  differentials  of  the  following : 

{a)    //  =  loge(^). 

ydx-xdy 
Ans.du=          ^^ 

{b)    z^  =  arctan^^). 

xdy-ydx 
Ans.  du-     ^,^^,    . 

{c)    u  =  xy. 

Ans.  du  =  xy-\ydxj 

38.  Total  Derivative.  We  may  in  (57)  assume  that  x 
and  J/  are  not  independent,  but  are  functions  one  of  the 
other,  say  J/  a  function  of  x.  Then  ti  becomes  also  a  func- 
tion  of  X  alone,  and   we   may  therefore   form   the   total 

derivative  — 
dx 

Dividing  (S/)  by  A;i:  and  taking  the  limit  for  t^x  =  o,  and 

.*.  Aj/  =  o,  we  have  the  result 

/  jj  N  ^  — .  ^  4.  fdzi\  dy 

dx      dx      \dyj  dx 

a  very  important  formula. 

Suppose  in  the  illustration  of  the  rectangle,  §  36,  we  wish  the  deriva- 
tive with  respect  to  the  base  x  of  the  area  u  of  all  rectangles  whose 
altitude/  is  double  the  base.     Then 

dx  dy  dx 

and  (59)  gives  ^=y  -\-  2x=  ax. 

dx 


PARTIAL   DERIVATIVES  89 

Or,  we  may  substitute  for/  before  differentiation ; 
i,e.  II  —  X'lx  —  2 x\   .'.  -—  =  4.Xj  as  before. 

Equation  (59)  is  especially  important  as  affording  a  proof  of  the 
method  given  in  §  19.     For  in  the  example  of  that  article,  set 

/^  =  X'  -  3^7  +  2/-  -  3; 
....  =  0,   and    -=.0,   or   -  +  ^-^=0; 


i,c.    (60) 


Qx     \dyldx 

du 
^___6f _        2x-2,y    __2x-2>y 
dx~m    du~      -3^+4J'~3^-4/' 


the  same  answer  as  before.     This  formula  (60)  is  very  useful. 
For  further  study  of  the  Calculus  the  student  is  referred  to : 

G.  A.  Gibson,  An  Elementary  Treatise  on  the  Calculus.    London, 

190 1. 
Young  and  Linebarger,   The  Elements  of  the  Differential  and 

Integral  Calculus.     New  York,  1900. 
McMahon  and   Snyder,  Elements  of  the  Differential  Calculus. 

New  York,  1898. 
Murray.     An  Elementary  Course  in  the  Integral  Calculus.     New 

York,  1898. 


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important  than  mere  simplicity ;  and  thus  it  is  hoped 
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and  at  the  same  time  useful  as  an  introduction  to  a  mcTre 
advanced  course  for  those  who  may  wish  to  specialize 
later  in  Mathematics. 


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Text-Books  on  Surveying 


RAYMOND'S  PLANE  SURVEYING 

By  William  G.  Raymond,  C.E.,  Member  American  Society 
of  Civil  Engineers  ;  Professor  of  Geodesy,  Road  Engineer- 
ing, and  Topograf)hical  Drawing  in  Rensselaer  Polytechnic 
Institute $3.00 

This  work  has  been  prepared  as^a  manual  for  the 
study  and  practice  of  surveying.  The  long  experience  of 
the  author  as  a  teacher  in  a  leading  technical  school  and 
as  a  practicing  engineer  has  enabled  him  to  make  the 
subject  clear  and  comprehensible  for  the  student  and 
young  practitioner.  It  is  in  every  respect  a  book  of 
modern  methods,  logical  in  its  arrangement,  concise  in  its 
statements,  and  definite  in  its  directions.  In  addition  to 
the  matter  usual  to  a  full  treatment  of  Land,  Topograph- 
ical, Hydrographical,  and  Mine  Surveying,  particular 
attention  is  given  to  system  in  office  work,  to  labor-saving 
devices,  the  planimeter,  slide  rule,  diagrams,  etc.,  to  co- 
ordinate methods,  and  to  clearing  up  the  practical  diffi- 
culties encountered  by  the  young  surveyor.  An  appendix 
gives  a  large  number  of  original  problems  and  illustrative 
examples. 

Other  Text-Books  on  Surveying 

DAVIES'S  ELEMENTS  OF  SURVEYING  (Van  Amringe)  .  .  $1.75 
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Edited  by  JOSEPH  S.  AMES,  Ph.D. 
Johns  Hopkins  University 


The  Free  Expansion  of  Gases.     Memoirs  by  Gay-Lussac,  Joule, 

and  Joule  and  Thomson.     Edited  by  Dr.  J.  S.  Ames    .         .    $0.75 

Prismatic    and    Diffraction    Spectra.      Memoirs    by   Joseph   von 

Fraunhofer.     Edited  by  Dr.  J.  S.  Ames 60 

Rbntgen  Rays.    Memoirs  by  Rontgen,  Stokes,  and  J.  J.  Thomson. 

Edited  by  Dr.  George  F.  Barker 60 

The  Modern  Theory  of  Solution.     Memoirs  by  Pfeffer,Van't  Hoff, 

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The  Laws  of  Gases.     Memoirs  by  Boyle  and  Amagat.     Edited  by 

Dr.  Carl  Barus 75 

The    Second    Law   of   Thermodynamics.      Memoirs   by  Carnot, 

Clausius,  and  Thomson.     Edited  by  Dr.  W.   F.   Magie     .        .90 

The  Fundamental  Laws  of  Electrolytic  Conduction.  Memoirs  by 
Faraday,  Hittorf,  and  Kohlrausch.  Edited  by  Dr.  H.  M. 
Goodwin 75 

The   Effects  of  a   Magnetic    Field  on   Radiation.      Memoirs   by 

Faraday,  Kerr,  and  Zeeman.     Edited  by  Dr.  E.  P.  Lewis  .        .75 

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Cavendish.     Edited  by  Dr.  A.  S.  Mackenzie      .         .         .      1  00 

The  Wave  Theory  of  Light.     Memoirs  by  Hu3'gens,  Young,  and 

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The  Discovery  of  Induced  Electric  Currents.     Vol.  H.     Memoirs 

by  Michael  Faraday.     Edited  by  Dr.  J.  S.  Ames  ...        .75 

Stereochemistry.  Memoirs  by  Pasteur,  Le  Bel,  and  Van't  Hoff, 
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and  others.     Edited  by  Dr.  G.  M.  Richardson  ,         .         .1.00 

The  Expansion  of  Gases.     Memoirs  by  Gay-Lussac  and  Regnault, 

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Biology  and  Zoology 


DODGE'S    INTRODUCTION    TO     ELEMENTARY     PRACTICAL 
BIOLOGY 

A  Laboratory  Guide  for  High  School  and  College  Students. 
By  Charles  Wright  Dodge,  M.S.,  Professor  of  Biology 

in  the  University  of  Rochester $1 .80 

This  is  a  manual  for  laboratory  work  rather  than  a 
text-book  of  instruction.  It  is  intended  to  develop  in  the 
student  the  power  of  independent  investigation  and  to 
teach  him  to  observe  correctly,  to  draw  proper  conclusions 
from  the  facts  observed,  to  express  in  writing  or  by  means 
of  drawings  the  results  obtained.  The  work  consists 
essentially  of  a  series  of  questions  and  experiments  on 
the  structure  and  physiology  of  common  animals  and 
plants  typical  of  their  kind — questions  which  can  be 
answered  only  by  actual  investigation  or  by  experiment. 
Directions  are  given  for  the  collection  of  specimens,  for 
their  preservation,  and  for  preparing  them  for  examination; 
also  for  performing  simple  physiological  experiments. 

ORTON'S     COMPARATIVE     ZOOLOGY,     STRUCTURAL     AND 
SYSTEMATIC 

By  James  Orton,  A.M.,  Ph.D.,  late  Professor  of  Natural 
History  "  in  Vassar  College.  New  Edition  revised  by 
Charles  Wright  Dodge,  M.S.,  Professor  of  Biology  in 

the  University  of  Rochester $1.80 

This  work  is  designed  primarily  as  a  manual  of 
instruction  for  use  in  higher  schools  and  colleges.  It 
aims  to  present  clearly  the  latest  established  facts  and 
principles  of  the  science.  Its  distinctive  character  con- 
sists in  the  treatment  of  the  whole  animal  kingdom  as  a 
unit  and  in  the  comparative  study  of  the  development  and 
variations  of  the  different  species,  their  organs,  functions, 
etc.  The  book  has  been  thoroughly  revised  in  the  light 
of  the  most  recent  phases  of  the  science,  and  adapted  to 
the  laboratory  as  well  as  to  the  literary  method  of  teaching. 


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Civics  and  Economics 


ANDREWS'S  NEW  MANUAL  OF  THE  CONSTITUTION 

COCKER'S  GOVERNMENT  OF  THE   UNITED   STATES 

FORMAN'S  FIRST  LESSONS  IN  CIVICS 

GREGORY'S  NEW  POLITICAL  ECONOMY      . 

LAUGHLIN'S  STUDY  OF  POLITICAL   ECONOMY. 

LAUGHLIN'S  ELEMENTS  OF   POLITICAL   ECONOMY 

McCLEARY'S  STUDIES  IN   CIVICS 

NORDHOFF'S  POLITICS  FOR  YOUNG  AMERICANS 
Revised  Edition  ...... 


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STORY'S    EXPOSITION    OF     THE     CONSTITUTION     OF    THE 

UNITED  STATES 90 

TOWNSEND'S  SHORTER  COURSE  IN   CIVIL  GOVERNMENT  .72 

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WILLOUGHBY'S  RIGHTS  AND  DUTIES  OF  AMERICAN  CITIZEN- 


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Text-Books  in  Geology 

By  JAMES  D.  DANA,  LL.D. 
Late  Professor  of  Geology  and  Mineralogy  in  Yale  University, 

DANA'S  GEOLOGICAL  STORY  BRIEFLY  TOLD  .  .  .  $1.15 
A  new  and  revised  edition  of  this  popular  text-book  for  beginners  in 
the  study,  and  for  the  general  reader.  The  book  has  been  entirely 
rewritten,  and  improved  by  the  addition  of  many  new  illustrations  and 
interesting  descriptions  of  the  latest  phases  and  discoveries  of  the  science. 
In  contents  and  dress  it  is  an  attractive  volume,  well  suited  for  its  use. 

DANA'S  REVISED  TEXT-BOOK  OF  GEOLOGY  .  .  .  $1.40 
Fifth  Edition,  Revised  and  Enlarged.  Edited  by  William  North 
Rice,  Ph.D.,  LL.D.,  Professor  of  Geology  in  Wesleyan  University. 
This  is  the  standard  text-book  in  geology  for  high  school  and  elementary 
college  work.  While  the  general  and  distinctive  features  of  the  former 
work  have  been  preserved,  the  book  has  been  thoroughly  revised,  enlarged, 
and  improved.  As  now  published,  it  combines  the  results  of  the  life 
experience  and  observation  of  its  distinguished  author  with  the  latest 
discoveries  and  researches  in  the  science. 

DANA'S  MANUAL  OF  GEOLOGY $5.00 

Fourth  Revised  Edition.  This  great  work  is  a  complete  thesaurus  of 
the  principles,  methods,  and  details  of  the  science  of  geology  in  its 
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physiography,  orogeny,  and  epeirogeny,  biologic  evolution,  and  paleon- 
tology. It  is  not  only  a  text-book  for  the  college  student  but  a  hand- 
book for  the  professional  geologist.  The  book  was  first  issued  in  1862, 
a  second  edition  was  published  in  1874,  ^^^  ^  third  in  1880.  Later 
investigations  and  developments  in  the  science,  especially  in  the  geology 
of  North  America,  led  to  the  last  revision  of  the  work,  which  was  most 
thorough  and  complete.  This  last  revision,  making  the  work  substantially 
a  new  book,  was  performed  almost  exclusively  by  Dr.  Dana  himself,  and 
may  justly  be  regarded  as  the  crowning  work  of  his  life. 


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